INTRODUCTORY COURSE IN DIFFERENTIAL EQUATIONS INTRODUCTORY COUR S IN DIFFERENTIAL EQUATIONS FOR STUDENTS IN CLASSICAL AND ENGINEERING COLLEGES BY D. A. MURRAY, B.A., PH.D. FORMERLY SCHOLAR AND FELLOW OF JOHNS HOPKINS UNIVERSITY INSTRUCTOR IN MATHEMATICS IN CORNELL UNIVERSITY LONGMANS, GREEN, AND CO. LONDON AND BOMBAY 1897 COPYRIGHT, 1897, BY LONGMANS, GREEN, AND CO. iLL RIGHTS REBERVED. Typography by J. S. Cushing & Co., Norwood, Mass., U. S. A. PREFACE. THE aim of this work is to give a brief exposition of some of the devices employed in solving differential equations. The book presupposes only a knowledge of the fundamental formule of integration, and may be described as a chapter supplementary to the elementary works on the integral calculus. The needs of two classes of students, with whom the author has been brought into contact in the course of his experience as a teacher, have determined the character of the work. For the sake of students of physics and engineering who wish to use the subject as a tool, and have little time to devote to general theory, the theoretical explanations have been made as brief as is consistent with clearness and sound reasoning, and examples have been worked in full detail in almost every case. Practical applications have also been constantly kept in mind, and two special chapters dealing with geometrical and physical problems have been introduced. The other class for which the book is intended is that of students in the general courses in Arts and Science, who have more time to gratify any interest they may feel in this subject, and some of whom may be intending to proceed to the study *of the higher mathematics. For these students, notes have v vi PREFACE. been inserted in the latter part of the book. Some of the notes contain the demonstrations of theorems which are referred to, or partially proved, in the first part of the work. If these discussions were given in full in the latter place, they would probably tend to discourage a beginner. Accordingly, it has been thought better to delay the rigorous proof of several theorems until the student has acquired some degree of familiarity with the working of examples. Throughout the book are many historical and biographical notes, which it is hoped will prove interesting. In order that beginners may have a larger and better conception of the subject, it seemed right to point out to them some of the most important lines of development of the study of differential equations, and notes have been given which have this object in view. For this purpose, also, a few articles have been placed in the body of the text. These articles refer to Riccati's, Bessel's, Legendre's, Laplace's, and Poisson's equations, and the equation of the hypergeometric series, which are forms that properly lie beyond the scope of an introductory work. In many cases in which points are discussed in the brief manner necessary in a work of this kind, references are given where fuller explanations and further developments may be found. These references are made, whenever possible, to sources easily accessible to an ordinary student, and especially to the standard treatises, in English, of Boole, Forsyth, and Johnson. For students who can afford but a minimum of time foi this study, the essential articles of a short course are indicated. after the table of contents. PREFACE. vii Of the examples not a few are original, and many are taken from examination papers of leading universities. There is also a large number of examples, which, either by reason of their frequent use in mechanical problems or their excellence as examples per se, are common to all elementary text-books on differential equations. There remains the pleasant duty of making confession of my indebtedness. In preparing this book, I have consulted many works and memoirs; and, in particular, have derived especial help for the principal part of the work from the treatises of Boole, Forsyth, and Johnson, and from the chapters on Differential Equations in the works of De Morgan, Moigno, Hoiel, Laurent, Boussinesq, and Mansion. I have in addition to acknowledge suggestions received from Byerly's " Key to the Solution of Differential Equations " published in his Integral Calculus, Osborne's Examples and Rules, and from the treatises of Williamson, Edwards, and Stegemann on the Calculus. Use has also been made of notes of a course of lectures delivered by Professor David Hilbert at Gôttingen. Suggestions and material for many of the historical and other notes have also been received from the works of Craig, Jordan, Picard, Goursat, Koenigsberger, and Schlesinger on Differential Equations; from Byerly's Fourier's Series and Spherical Harmonics, Cajori's History of lMathematics, and from the chapters on Hyperbolic Functions, Harmonic Functions, and the History of Modern Mathematics in Merriman and Woodward's Iigher Mathematics. The mechanical and physical examples have been obtained from Tait and Steele's Dynamics of a Particle, Ziwet's Mechanics, Thomson and Tait's Natural Philosophy, viii PREFA CE. Eimtage's Mathematical Theory of Electricity and Magnetism, Bedell and Crehore's Alternating currents, and Bedell's Principles of the Transformer. These and many other acknowledgments will be found in various parts of the book. To the friends who have encouraged and aided me in this undertaking, I take this opportunity of expressing my gratitude. And first and especially to Professor James McMalhon of Cornell University, whose opinions, advice, and criticisms, kindly and freely given, have been of the greatest service to me. I have also to thank Professors E. Merritt and F. Bedell of the department -of physics, and Professor Tanner, Mr. Saurel, and Mr. Allen of the department of mathematics at Cornell for valuable aid and suggestions. Professor McMahon and Mr. Allen have also assisted me in revising the proof-sheets while the work was going through the press. To Miss H. S. Poole and Mr. M. Macneill, graduate students at Cornell, I am indebted for the verification of many of the examples. D. A. MURRAY. CORNELL UNIVERSITY, April, 1897. CONTENTS. EQUATIONS INVOLVING TWO VARIABLES. CHAPTER I. DEFINITIONS. FORMATION OF A DIFFERENTIAL EQUATION. ART. PAGE 1. Ordinary and partial differential equations. Order and degree 1 2. Solutions and constants of integration.. 2 3. The derivation of a differential equation... 4 4. Solutions, general, particular, singular.. 6 5. Geometrical meaning of a differential equation of the first order and degree..... 8 6. Geometrical meaning of a differential equation of a degree or an order higher than the first.. 9 Examples on Chapter I. 11 CHAPTER II. EQUATIONS OF THE FIRST ORDER AND OF THE FIRST DEGREE. 8. Equations of the form fi(x)dx + f2(y)dy = O.. 14 9. Equations homogeneous in x and y...... 15 10. Non-homogeneous equations of the first degree in x and y. 16 11. Exact differential equations....... 17 12. Condition that an equation of the first order be exact.. 18 13. Rule for finding the solution of an exact differential equation. 19 14. Integrating factors......... 21 15. The number of integrating factors is infinite.... 21 16. Integrating factors found by inspection.... 22 17. Rules for finding integrating factors. Rules I. and II... 23 18. Rules III. and IV..... 24 ix X %CONTENTS. ART. PAGE 19. Rule V........... 25 20. Linear equations.... 26 21. Equations reducible to the linear form..... 28 Examples on Chapter II..... 29 CHAPTER III. EQUATIONS OF THE FIRST ORDER BUT NOT OF THE FIRST DEGREE. 22. Equations that can be resolved into component equations of the first degree..... 31 23. Equations that cannot be resolved into component equations. 32 24. Equations solvable for y........ 33 25. Equations solvable for x...... 34 26. Equations that do not contain x; that do not contain y..34 27. Equations homogeneous in x and y......35 28. Equations of the first degree in x and y. Clairaut's equation. 36 29. Summary...... 38 Examples on Chapter III........ 38 CHAPTER IV. SINGULAR SOLUTIONS. 30. References to algebra and geometry... 40 31. The discriminant......... 40 32. The envelope.... 41 33. The singular solution... 42 34. Clairaut's equation........ 44 35. Relations, not solutions, that may appear in the p and c discriminant relations... 44 36. Equation of the tac-locus.... 45 37. Equation of the nodal locus.. 45 38. Equation of the cuspidal locus... 47 39. Summary...... 48 Examples on Chapter IV.... 49 CHAPTER V. APPLICATIONS TO GEOMETRY, MECHANICS, AND PHYSICS. 41. Geometrical problems........ 51 42. Geometrical data........ 51 CONTENTS. xi ART. PAGE 43. Examples...... 53 44. Problems relating to trajectories...... 55 45. Trajectories, rectangular co-ordinates..... 55 46. Orthogonal trajectories, polar co-ordinates. 56 47. Examples...... 57 48. Mechanical and physical problems..... 58 Examples on Chapter V...60 CHAPTER VI. LINEAR EQUATIONS WITI CONSTANT COEFFICIENTS. 49. Linear equations defined. The complementary function, the particular integral, the complete integral... 63 50. The linear equation with constant coefficients and second member zero..... 64 51. Case of the auxiliary equation having equal roots... 65 52. Case of the auxiliary equation having imaginary roots.. 53. The symbol D..... 67 54. Theorem concerning D..... 68 55. Another way of finding the solution when the auxiliary equation has repeated roots..... 69 56. The linear equation with constant coefficients and second member a function of x..... 70 57. The symbolic function.... 70 f(D) 58. Methods of finding the particular integral... 72 59. Short methods of finding the particular integrals in certain cases....... 60. Integral corresponding to a term of form eax in the second member.......... 74 61. Integral corresponding to a term of form x1n in the second member.......... 75 62. Integral corresponding to a term of form sin ax or cos ax in the second member.... 76 63. Integral corresponding to a term of form eax T in the second member.......... 78 64. Integral corresponding to a term of form xV in the second member..........79 Examples on Chapter VI.,...80 xii CONTENTS. CHAPTER VII. LINEAR EQUATIONS WITII VARIABLE COEFFICIENTS. ART. PAGE 65. The homogeneous linear equation. First method of solution. 82 66. Second method of solution: (A) To find the complementary function.......... 84 67. Second method of solution: (B) To find the particular integral 85 68. The symbolic functions f() and... 86 f(O) 69. Methods of finding the particular integral....87 70. Integral corresponding to a term of form xa in the second member..........89 71. Equations reducible to the homogeneous linear form.. 90 Examples on Chapter VII.... 91 CHAPTER VIII. EXACT DIFFERENTIAL EQUATIONS AND EQUATIONS OF PARTICULAR FORnIS. INTEGRATION IN SERIES. 73. Exact differential equations defined......92 74. Criterion of an exact differential equation... 92 75. The integration of an exact equation; first integrals.. 94 76. Equations of the form d-Y-f(x).....96 clx" 77. Equations of the form d =f(Y)... 96 78. Equations that do not contain y directly.. 97 79. Equations that do not contain x directly..... 98 80. Equations in which y appears in only two derivatives whose orders differ by two... 99 81. Equations in which y appears in only two derivatives whose orders differ by unity....100 82. Integration of linear equations in series..... 101 83. Equations of Legendre, Bessel, Riccati, and the hypergeometric series......105 Examples on Chapter VIII.... 107 CONTENTS. Xiii CHAPTER IX. EQUATIONS OF THE SECOND ORDER. ART. PAGE 85. The complete solution in terms of a known integral... 109 86. Relation between the integrals.......111 87. To find the solution by inspection......111 88. The solution found by means of operational factors.. 112 89. The solution found by means of two first integrals.. 114 90. Transformation of the equation by changing the dependent variable..........114 91. Removal of the first derivative.......115 92. Transformation of the equation by changing the independent variable.....117 93. Synopsis of methods of solving equations of the second order. 118 Examples on Chapter IX.. 120 CHAPTER X. GEOMETRICAL, MECHANICAL, AND PHYSICAL APPLICATIONS. 95. Geometrical problems........121 96. Mechanical and physical problems... 122 Examples on Chapter X. 124 EQUATIONS INVOLVING MORE THAN TWO VARIABLES. CHAPTER XI. ORDINARY DIFFERENTIAL EQUATIONS WITH MORE THAN Two VARIABLES. 98. Simultaneous differential equations which are linear..128 99. Simultaneous equations of the first order.... 130 100. General expression for the integrals of simultaneous equations of the first order.....133 101. Geometrical Ineaning of simultaneous differential equations of the first order and the first degree involving three variables. 134 102. Single differential equations that are integrable. Condition of integrability.. 136 103. Method of finding the solution of the single integrable equation 137 xiv CONTENTS. ART. PAGE 104. Geometrical meaning of the single differential equation which is integrable.... 140 105. The locus of Pdx + Qdy + Rdz = 0 is orthogonal to the locus of dx dy = dz 141 P Q R 106. The single differential equation which is non-integrable.. 142 Examples on Chapter XI. 143 CHAPTER XII. PARTIAL DIFFERENTIAL EQUATIONS. 107. Definitions.......... 146 108. Derivation of a partial differential equation by the elimination of constants..... 146 109. Derivation of a partial differential equation by the elimination of arbitrary functions.. 148 PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 110. The integrals of the non-linear equation: the complete and particular integrals.... 149 111. The singular integral........150 112. The general integral........150 113. The integral of the linear equation..... 153 114. Equation equivalent to the linear equation...154 115. Lagrange's solution of the linear equation....154 116. Verification of Lagrange's solution...... 155 117. The linear equation involving more than two independent variables..........156 118. Geometrical meaning of the linear partial differential equation 158 119. Special methods of solution applicable to certain standard forms. Standard I.: equations of the form F(p, q)= 0. 159 120. Standard II.: equations of the form z =-)X + qy + f (1, q) 161 121. Standard III.: equations of the form F(z, p, q) = 0.. 162 122. Standard IV.: equations of the form fl(x,p) =ff(y, q). 164 123. General method of solution..... 166 PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND AND HIGHER ORDERS. 124. Partial equations of the second order..... 169 125. Examples readily solvable...... 170 CONTENTS. xv ART. PAGE 126. General method of solving Rr + Ss + Tt = V... 171 127. The general linear partial equation of an order higher than the first.......173 128. The homogeneous equation with constant coefficients: the complementary function.......174 129. Solution when the auxiliary equation has repeated or imaginary roots......175 130. The particular integral..... 176 131. The non-homogeneous equation with constant coefficients: the complementary function.......179 132. The particular integral.......180 133. Transformation of equations....... 182 134. Laplace's equation, V2V- 0....... 182 135. Special cases......... 185 136. Poisson's equation, V2V - 4 rp...... 186 Examples on Chapter XII..187 MISCELLANEOUS NOTES. A. Reduction of equations to a system of simultaneous equations of the first order........ 189 B. The existence theorem.. 190 C. The number of constants of integration..... 194 D. Criterion for the independence of constants... 195 E. Criterion for an exact differential equation. 197 F. Criterion for the linear independence of the integrals of a linear equation.......... 197 G. Relations between the integrals and the coefficients of a linear equation.......... 199 H. Criterion of integrability of Pdx + Qdy + Rdz = 0... 200 I. Modern theories of differential equations. Invariants.. 202 J. Short list of works on differential equations... 205 Answers to the examples... 209 SHORT COURSE. (The Roman numerals refer to chapters, the Arabic to articles.) I.; II. 7-16, 20, 21; III.; IV. 30-34; V.; VI. 49-53, 56-62; VII. 65, 66, 71; VIII. 72-81; IX. 84, 85, 87, 90-93; X.; XI. 97-99, 101 -103, 106; XII. 107-116, 119-122, 124, 125, 127, 128, 131, 133. DIFFERENTIAL EQUATIONS. CHAPTER I. DEFINITIONS. FORMATION OF A DIFFERENTIAL EQUATION. 1. Ordinary and partial differential equations. Order and degree. A differential equation is an equation that involves differentials or differential coefficients. Ordinary differential equations are those in which all the differential coefficients have reference to a single independent variable. Thus, dy = cos xdx, (1) = 0, (2) d2y (2y + C)(~+ z - (y + ) = 0, (3) Y~d" + + (d) ( ) dx2 Y = + d- (6) dx dy are ordinary y differential equations. B 1 2 DIFFERENTIAL EQUATIONS. [CH. I. Partial differential equations are those in which there are two or more independent variables and partial differential coefficients with reference to any of them; as, y + xy-=nxz. dx dy The order of a differential equation is the order of the highest derivative appearing in it. The degree of an equation is the degree of that highest derivative, when the differential coefficients are free from radicals and fractions. Of the examples above, (1) is of the first order and first degree, (2) is of the second order and first degree, (4) is of the first order and second degree, (5) is of the second order and second degree, (6) is of the first order and second degree. In the integral calculus a very simple class of differential equations of which (1) is an example have been treated. Equations having one dependent variable y and one independent variable x will first be considered. The typical form of such equations is f dy dn.y. f( ~x, dx~! — O=. 2. Solutions and constants of integration. Whether a differential equation has a solution, what are the conditions under which it will have a solution of a particular character, and other questions arising in the general theory of the subject are hardly matters for an introductory course.* The student will remember that he solved algebraic equations, before he could prove that such equations must have roots, or before he had more than a very limited knowledge of their general properties. This book will be concerned merely with an exposition of the methods of solving some particular classes of differential equations; and their solutions will be expressed by the ordinary algebraic, trigonometric, and exponential functions. * For a proof that a differential equation has an integral, and for references relating to this fundamental theorem, see Note B, p. 190. ~ 2.] CONSTANTS OF INTEGRATION. 3 A solution or integral of a differential equation is a relation between the variables, by means of which and the derivatives obtained therefrom, the equation is satisfied. Thus y = sin x is a solution of (1); x2 + y2 = r2 and y = mx + rl- + m2 are solutions of (4) Art. 1. In two of these solutions, y is expressed explicitly in terms of x, but in the solutions of differential equations in general, the relation between x and y is oftentimes not so simply expressed. This will be seen by glancing at the solutions of the examples on Chapter II. A solution of (1) Art. 1 is y sin x; another solution is y = sin x + c, (1) c being any constant. By changing the value of c, different solutions are obtained, and in particular, by giving c the value zero, the solution y = sin x is obtained. A solution of dx2 + = 0 (2) is y = sin x, and another solution is y = cos x. A solution more general than either of the former is y = A sin x; and it includes one of them, as is seen by giving A the particular value unity. Similarly y = B cos x includes one of the two first given solutions of (2). The relation y = A cos x + B sin x (3) is a yet more general solution, front which all the preceding solutions of (2) are obtainable by giving particular values to A and B. The arbitrary constants A, B, c, appearing in these solutions are called arbitrary constants of integration. Solution (1) has one arbitrary constant, and solution (3) has two; the question arises: IHow many arbitrary constants must the most general solution of a differential equation contain? 4 DIFFERENTIAL EQUATIONS. [CH. I. The answer can in part be inferred from the consideration of the formation of a differential equation. 3.* The derivation of a differential equation. In the process of deriving (2) from (3) Art. 2, A and B have been made to disappear. To eliminate two constants, A and B, three equations are required. Of these three equations, one is given, namely, (3), and the two others needed are obtained by successive differentiation of (3). Thus, y A=sinx Bcosx, d = A cos x - Bsi x, dx. d2y dx2 - A sin x - B cos x; d2Y whence,d l+ y=~0. Now consider the general process. The equation f (X, C1, C2,..., c,)= O (1) contains, besides x anc y, n arbitrary constants c,2,...,c,, Differentiation n times in succession with respect to x gives df cdf dy dx dy dx d + 2 C2 dy d2 + f cly d?\2 + f C o2 de d clx dy dcx dykclx) ~y da2 dnf cf d"y O d ' ' ' dy dx Between the original equation and the n equations thus obtained by differentiation, making n + 1 equations ini all, the * See B. Williamson, Differential Calculus, Art. 311; J. Edwards, Differential Calculus, Arts. 506, 507. ~ 3.] FORMATION OF A DIFFERENTIAL EQUATION. 5 n constants cc, c*2,, cana be eliminated, and thus will be formed the equation a,dy,... 2y d"y\ (2) F[ Y3 dx' dx2 - )=( Therefore, when there is a relation between x and y involving n arbitrary constants, the corresponding differential relation which does not contain the constants is obtained by eliminating these n constants froni the n + 1 equations, made up of the given relation and n new equations arising from n successive differentiations. There being n differentiations, the resulting equation must contain a derivative of thle nth order, and therefore a relation between x and y, involving n arbitrary constants, will give rise to a differential equation of the nth order free froin those constants. The equation obtained is independent of the order in which, and of the manner in which, the eliminations are effected.* On the other hand, it is evident that a differential equation of the nth order cannot have more than n arbitrary constants in its solution; for, if it had, say n + 1, on eliminating them there would appear, not an equation of the nth order, but one of the (n + 1)th order.t Ex. 1. From x2 + y2 + 2 ax + 2 by + c = 0, derive a differential equation not containing a, b, or c. Differentiation tree times in succession gives _dy dly x +- Y+ta+ b = 0, ^dy cy (2 ^y 2y 1 - + y + b = dy d'2y y d3y __Y 0. dx c-X2 i+ yd3+ -bX3 * See Joseph Edwards, Digferential Calculus, Art. 507, after reading Arts. 5, 6, following. t For a proof that the general solution of an equation of the nth order contains exactly n arbitrary constants, see Note C, p. 194. 6 DIFFERENTIAL EQUATIONS. [CH. I. The elimination of b froin the last two equations gives the differential equation required, (lx dx3 dx (dx2' Ex. 2. Form the differential equation corresponding to y2 - 2 ay + X2 = a2, by eliminating a. Ex. 3. Eliminate a and p from (x - a)2 + (y- P)2 = r2. Ex. 4. Eliminate m and a from y m (a2 - x2). 4. Solutions, general, particular, singular. The solution which contains a number of arbitrary constants equal to the order of the equation, is called the general solution or the complete integral. Solutions obtained therefrom, by giving particular values to the constants, are called particular solutions. Looking on the differential equation as derived from the general solution, the latter is called the complete primitive of the former. It nay be noted that from the relation (1) Art. 3 several differential equations can be derived, which are different when the constants chosen to be eliminated are different. Thus, the elimination of all the constants gives but one differential equation, namely (2), for the order of elimination does not affect the equation formed. The elimination of all but cl gives an equation of the (n - 1)th order; elimination of all but c, gives another equation of the (n - 1)th order; anc similarly for c3,..., c^. So from (1), n equations of the (n - 1)th order can be derived. Therefore (1) is the complete primitive of one equation of the nth order, and the complete primitive of n different equations of the (n - 1)th order. The student may determine how many equations of the first, second,..., (n - 2)th order can be derived from (1). The general solution may not include all possible solutions. For instance, (4) Art. 1 has for solutions, x2 + y2 = r2, and y = mx + rV/ + nm. The latter is the general solution, containing the arbitrary constant m, but the former is not deriva ~ 4.] SOLUTIONS. 7 ble from it by giving particular values to m. It is called a singular solution. Singular solutions are discussed in Chapter IV. The n arbitrary constants in the general solution must be independent and not equivalent to less than n constants. The solution y = cex+x appears to contain two arbitrary constants c and a, but they are really equivalent to only one, for y + ce+a- = cede- = Aex, and by giving A all possible values, all the particular solutions, that can be obtained by giving c and a all possible values, will also be obtained.* The general solution cau have various forms, but there will be a relation. between the arbitrary constants of one forln and those of another. For example, it bas been seen that the gend2yl eral solution of - y = 0 is y =A sin x + B cos x. But y = c sin (x + a) is also a solution, as may be seen by substitution in the given equation; and it is a general solution, since it contains two independent constants c and a. The latter form expanded is y = c cos a sin x + c sin a cosx. On comparing this form with the first form of solution given, it is evident that the relations between the constants A, B, of the first form and c, a, of the second, are A = c cos a, and B = c sin a, that is, B c = VA2 + B2, and a = tan-1. If the solution has to satisfy other conditions besides that made by the given differential equation, some or all of the constants will have determinate values, according to the number of conditions imposed. ~ See Note D for a criterion of the independence of the constants. 8 DIFFERENTIAL EQUATIONS. [CH. I. 5. Geometrical meaning of a differential equation of the first order and degree. Take f(x, Yj =o, (1) an equation of the first degree in d. It will be remembered, that when the equation of a curve is given in rectangular co-ordinates, the tangent of its direction at any point is dy For any particular point (xz, y,), there will be a corresponding particular value of dy say m i, determined by equation (1). A idx point that moves, subject to the restriction imposed by this equation, on passing through (x,, y,) must go in the direction nmi. Suppose it moves from (x, y,) in the direction m, for an infinitesimal distance, to a point (x2, y2); then, that it moves from (x2, y,) in the direction mn, the particular direction associated with (x,, y2) by the equation, for an infinitesimal distance to a point (x,, y3); thence, under the same conditions to (x4, Y4), and so on through successive points. In proceeding thus, the point will describe a curve, the co-ordinates of every point of which, and the direction of the tangent thereat, will satisfy the differential equation. If the moving point starts at any other point, not on the curve already described, and proceeds as before, it will describe another curve, the co-ordinates of whose points and the direction of the tangents thereat satisfy the equation. Through every point on the plane, there will pass c/y a particular curve, for every point of which, x, y, will satisfy the equation. The equation of each curve is thus a particular solution of the differential equation; the equation of the system of such curves is the general solution; and all the curves represented by the general solution, taken together, make the locus of the differential equation. There being one arbitrary constant in the general solution of an equation of the first order, the locus of the latter is made up of a single infinity of curves, ~ 5, 6.] GEOMETRICAL MEANING. 9 Ex. 1. The equation y clx y indicates that a point moving so as to satisfy this equation, moves perpendicularly to the line joining it to the origin; that is, it describes a circle about the origin as centre. Putting the equation in the form x x + J dy = O, it is seen that the general solution is x2 + y2 = c. The circle passing through a particular point, as (3, 4), is x2 + y2 = 25, which is a particular solution. The general solution thus represents the system of circles having the origin for centre, and the equation of each one of these circles is a particular solution. That is, the locus of the differential equation is made up of all the circles, infinite in number, that have the origin for centre. Ex. 2. x dy + y cx = O has for its solution, xy = c, the equation of the system of hyperbolas, infinite in number, that have the x and y axes for asymptotes. dx having for its solution, y -= mx + c, las for its locus all straight lines, infinite in number, of slope m. 6. Geometrical meaning of a differential equation of a degree or an order higher than the first. If 7( =o If f Y' dy) = is of the second degree in dy, there will be two values of dy dx dx belonging to each particular point (xi, y,). Therefore the moving point can pass through each point of the plane in either of 10 DIFFERENTIAL EQUATIONS. [CH. I. two directions; and hence, two curves of the system which is the locus of the general solution pass through each point. The general solution, (x y, c)= 0, must therefore have two different values of c for each point; and hence, c must appear in that solution in the second degree. In general, it may be said: A differential equation, (". Yd- = 0, which is of the nth degree in dy, and which has dxc 4 (x, y, c)= o for its general solution,.has for its locus a single infinity of curves, there being but one arbitrary constant in >; n of these curves pass through each point of the plane, since dy has n clx values at any point; and hence the constant c must appear in the nth degree in the general solution. The general solution of a differential equation of the second order, f(, y dy d2y )_O k'^ i'dx 2) contains two arbitrary constants, and will therefore have for its locus a double infinity of curves; that is, a set of curves oo2 in number. Ex. d2 0 dx2 has for its solution, y =: mx + c, m and c being arbitrary. A line through any point (0, c), drawn in any direction m, is the locus of a particular integral of the equation. On taking a particular value of c, say Ci, there will be an infinity of lines corresponding to the infinity of values that m can have, and all these lines are loci of integrals. Since to each of the infinity of values that c can have there corresponds an infinity of lines, the complete integral will represent a doubly infinite system ~ 6.] EXAMPLES. 11 of straight lines; in other words, the locus of that differential equation consists of a doubly infinite system of lines. This can be deduced from other considerations. The condition d2y d 2 =0 requires, and requires only, that the curve described by the moving point shall have zero curvature, that is, it can be any straight line; and there can be oo2 straight lines drawn on a plane. Ex. 2. All circles of radius r, mo2 in number, are represented by the equation (x - a)2 + (y - b)2 = r,2 where a and b, the co-ordinates of the centre, are arbitrary. On eliiiinating a and b, there appears {i dy) 2 d2Y \ (ldxl r dx2 Thus, the locus of the latter equation of the second order consists of the doubly-infinite system of circles of radius r. Ex. 3. The locus of the differential equation of the third order, derived in Example 1, Art. 3, includes all circles, oo3 ir number; for it is derived from a complete primitive which has a, b, c arbitrary and thus represents circles whose centres and radii are arbitrary. It will have been observed from the above examples on lines and circles, that as the order of the differential equation rises, its locus assumes a more general character. EXAMPLES ON CHAPTER I. 1. Eliminate the constant a from /1 - x2 + V1 - y2 = a (x - y). 2. Form the differential equation of which y = cesin-'x is the complete integral. 3. Find the differential equation corresponding to y = ae2S + be-3X + cex, where a, b, c are arbitrary constants. 4. Form the differential equation of which c (y + c)2 = x3 is the complete integral. 5. Eliminate c from y = cx + c - c3. 12 DIFFERENTIAL EQUATIONS. [CH. I. 6. Eliminate c from ay2 = (x - c)3. 7. Form the differential equation of which e2y + 2 cxey + c2 = O is the complete integral. 8. Eliminate a and b from xy = aex + be-x. 9. Form the differential equation which has y-c a cos (mx + b) for its complete integral, a and b being the arbitrary constants. 10. Form the differential equation that represents all parabolas each of which has a latus;ectum 4 a, and wke-Xscres l~cLra-lie o-to- the- x 11. Find the differential equation of all circles which pass through the origin and wliose centres are on the x axis. 12. Form the differential equation of all parabolas whose axes are parallel to the axis of y. 13. Forin the differential equation of all conics whose axes coincide with the axes of co-ordinates. 14. Eliminate tle constants from y2 = ax + bx2. ~ 7.] FIRST ORDER AND FTRST DEGREE. 13 CHAPTER II. EQUATIONS OF THE FIRST ORDER AND OF THE FIRST DEGREE. 7. In Chapter I. it has been shown how to deduce from a given relation between x, y, and constants, a relation between x, y, and the derivatives of y with respect to x. There has now to be considered the inverse problem: viz., from a given relation between x, y, and the derivatives of y, to find a relation between the variables themselves. As, for instance, the problem of finding the roots of an algebraic equation is more difficult than that of forming the equation when the roots are given; or as, again, integration is a more difficult process than differentiation; so here, as in other inverse processes, the process of solving a differential equation is much more complicated and laborious than the direct operation of forming the equation when the general solution is given. An equation is said to be solved, when its solution has been reduced to expressions of the forms ff(x)dx, j P(y)dy, even if it be impossible to evaluate these integrals in terms of known functions. The equation f(x, y, '','... = 0 cannot be solved in Thie equairtion f dx, _'dy, dyl every case. In fact, even Pd + Q = O, where P and Q are dx functions of x and y, cannot be solved completely. It will be remembered how few in number are the solvable cases in algebraic equations; and it is the same with differential equations. The remainder of this book will be taken up with a 14 DIFFERENTIAL EQUATIONS. [CH. II. consideration of a few special forms of equations and the methods devised for their solution.* This chapter will be devoted to certain kinds of equations of the first order and degree, viz.: 1. Those that are either of the form f1(x) dx +f2(Y) dy = 0, or are easily reducible to this form; 2. Those that are reducible to this form by the use of special devices - (a) Equations homogeneous in x and y. (b) Non-homogeneous equations of the first degree in x and y; 3. Exact differential equations, and those that can be made exact by the use of integrating factors; 4. Linear equations and equations that are reducible to the linear form. 8. Equations of the form fi (x) dx + f2 () dy = 0. When an equation is in the form f (x) dx + f2 () dy = 0, its solution, obtainable at once by integration, is fi (x) dx + f2 (y) dy c. If the equation is not in the above form, sometimes one can see at a glance how to put it in that form, or, as it is commonly expressed, to separate the variables. Ex. 1. (1) (1-x)dy - (1 + y)dx = can evidently be written (2) dy dx -0, I+y 1-x * The student who is proceeding to find the methods of solving differential equations with no more knowledge of the subject than that imparted in the preceding pages, is reminded that he does this, ass'uming (1) that every differential equation with one independent variable has a solution, and (2) that this solution contains a number of arbitrary constants equal to the number indicating its order. ~ 8, 9.] OMOGENEOUS EQUATIONS. 15 whence, on integrating, (3) log(1 + y) + log(1 - x)= c, and hence (4) (1 + y)(1 - x)= e = ci. In equations (3) and (4) appear two ways of expressing a general solution of the same equation. Both are equally correct and equally general, but the one has the advantage over the other in neatness and simplicity, and this would make it more serviceable in applications. In some of the examples set, the reduction of the solutions to forms neater and simpler than those which at first present themselves, may require as much labour as the solving of the equations. The solution (4) could have been obtained without separating the variables, if one had noticed that (1 - x)dy -(1 + y)dx is the differential of (1 - x)(1 + y). Here, as in the calculus and other subjects, the experience that cores from practice, is the best teacher for showing how to work in the easiest way. Equation (1) can also be put in the forin dy - dx-(xdy + ydx)= O, and another form of the solution obtained, namely, y - - xy = C2. Solution (4) reduces to this form on putting c2 for ci - 1. Ex. 2. Solve + /- =0. dx dx] d dx Ex. 3. Solve (y-xi =a( 2y+~). Ex. 4. Solve 3 ex tan y dx + (1 - ex) sec2J y d= O. 9. Equations homogeneous in x and y. These equations can be put in the form dy fi (x, y) dx f (x, y)' where f, f2, are expressions homogeneous and of the same degree in x and y. On putting y = vx9 this equation becomes v+ x F(v), dIx 16 DIFFERENTIAL EQUATIONS. [CH. II. since each term ini f, f2, is of the same degree, say n, in x; and x'~ is thus a factor common to both numerator and denominator of its right-hand inember. Separation of the variables gives dv dx F(v)- v x' the solution of which gives the relation between x and v, that is, between y and Y, which satisfies the original equation. x Ex. 1. Solve (x2 + y2)dx - 2 xydy = 0. Putting y = vx gives (1 + v2)dx - 2 v(xdv + vdx)= O, which, on separation of the variables, reduces to dx 2 d dv -- O. x 1 - v2 Integrating, loo x(1 - 2) log c. On changing the logarithmic form to the exponential, and putting x for v, the solution becomes Y x2 _ y2 = cx. Ex. 2. Solve y2 dx + (xy + x2)dy = 0. Ex. 3. Solve x2ydx - (x3 + y3)dy = O. Ex. 4. Solve (4 y +- 3 x) d- y - 2 x = 0. (x, 10. Non-homogeneous equations of the first degree in x and y. These equations are of the forn dy _ x +- by +' c dx a'x + b'y +-c' For x put x' + h, and for y put y' - k, where h and k are constants; then dx = cx' and dy = dy', and (1) becomes dy' c_ ax' + by' + ah7 + bk + c dx' a'x'+ b'y' - a'h + b'k + c' If h and k are determined, so that ah + bk + c = O, ~ 10, 11.] EXACT DIFFERENTIAL EQUATIONS. 17 and a'h + b'k + c'= O, then (1) becomes dy x' + (2)b dx' ax' + b'y" which is homogeneous in x' and y', and therefore solvable by the method of Art. 6. If (2) has for its solution f(x', y')= O, the solution of (1) is f(x- h), (y - )= 0. This method fails when a: b = a': b', h and k then being infinite or indeterminate. Suppose ac b m1 then (1) can be written. cdy_ ax + by +c dcx m(cx + by) + c' On putting v for ax + by, the latter equation becomes cdv, v + c dx mv + c where the variables can be separated. Ex. 1. Solve (3 y-7 x+7)dx +(7y-3x + 3)dy=0. Ex. 2. Solve (y-3zx+3) - =2y-x-4. 11. Exact differential equations. A differential equation which has been formed from its primitive by differentiation, and without any further operation of elimination or reduction, is said to be exact; or, in other words, an exact differential equation is formed by equating an exact differential to zero. There has now to be found the condition which the coefficients of an equation must satisfy, in order that it may be exact, and also the method of solution to be employed when that condition is c 18 DIFFERENTIAL EQUATIONS. [CH. II. satisfied. The question of how to proceed when the condition is not satisfied will be considered next in order. 12. Condition that an equation of the first order be exact. What is the condition that Mdx + Ndy = 0 (1) be an exact differential equation, that is, that Mdx + Ndcy be an exact differential? In order that Mdx + Ndy be an exact differential, it must have been derived by differentiating some function u of x and y, and performing no other operation. That is, du = Mdx + Ndy. But du = adx + udy. Dx. y Hence, the conditions necessary, that Mdx + Ndy be the differential of a function u, is that M= 7, and N= (2) ax ôy The elimination of u imposes on M, N, a single condition, DM DN -y-= Dx' (3) ay =' OCu since each of these derivatives is equal to D.u Dx Dy This condition is also sufficient for the existence of a function that satisfies (1).* If there is a function u, whose differential du is such that du = Mdx + Ndy, then on integrating relatively to x, since the partial differential Mdx can have been derived only froin the terms containing x, u = -Mdx + terms not containing x, that is, u =fMdx + Fy) (4) * For another proof see Note E. ~ 12, 13.] EXACT DIFFERENTIAL EQUATIONS. 19 Differentiating both sides of (4) with respect to y, Ou _a,Mx d F(y) ay ayJ dy Ou But by (2), - must equal N, hence dF(y) = N f dx. (5) dy OJ The first member of (5) is independent of x; so, also, is the second; for differentiating it with respect to x gives -a-, dx ôy which, by condition (3), is zero. Integration of both sides of (5) with respect to y gives F(y)= N- f a Mdx dy a, where a is the arbitrary constant of integration. Substitution in (4) gives u= XxMdx + N-+ Mdx$ dy + a. J J ( ôS~ ayJ )~.`"( Therefore the primitive of (1), when condition (3) is satisfied, is Mdx + Nx - -adx ay = c. (6) Similarly, f Ndy + { M — + Ndy dx =_ c is also a solution. 13. Rule for finding the solution of an exact differential equation. Since all the terms of the solution that contain x must appear in -fMdx, the differential of this integral with respect to y must have all the terms of Ndy that contain. x; and therefore (6) can be expressed by the following rule: To find the solution of an exact differential equation, Mdx + Ndy = 0, integrate Mdx as if y were constant, integrate the terms in Nldy that do not contain x, and equate the suni of these integrals to a constant. 20 DIFFERENTIAL EQUATIONS. [C. II. Ex. 1. Solve (x2- 4 xy'-2 y2)dx + (y - 4 xy - 2 x2)dy = O. Here, -_ _ 4 x,- 4 y -= -; hence it is an exact equation. d d3 Mdx is 3- -2 x2y - 2 xy2; y2 dy is the only term iIn Ndy free from x. JBta iS3 Therefore the solution is xa Y3 -2 x2y -2x_ 2 X2 - = l, 3 3 or x3- 6 x2y -6 xy2 + y3 = c. The application of the test and of the rule can sometimes be simplified. By picking out the terms of llMdx + -Ndy that obviously fori an exact differential, or by observing, whether any of the ternis can take the forin Jlu)ldul, an expression less cumbersome than the original remains to be tested and integrated. For instance, the terms of the equation in this example can be rearranged thus: x2 dx + y2 dy - (4 xy + 2 y2)dx - (4 xy + 2 x2)dy = O. The first two terms are exact differentials, and the test has to be applied to the last two only. xdy - y (dx Ex. 2. xdx + ydy + x2 y-= 0 x2 + y0 becomes, on dividing the numerator and denominator of the last term by x2, xdx + ydy +.-, each term of which is an exact differential. Integrating, 2 + y2 + tan-l = c. 2 x Ex. 3. Solve (a - 2xy - y2)dx(x f y)2dy= 0.. Ex. 4. Solve (2 ax + by + g)dx + (2 cy + bx + e)dy = 0. Ex. 5. Solve (2 x2y + 4 x3- 12 xy2 + 3 y2 - xey + e2)dy +- (12 x2y 2 y2 +x2 4 x3 - 4y3 + 2 ye2 - e)dx = 0. ~ 14, 15.] INTEGRATING FACTORS. 21 14. Integrating factors. 'he differential equation ydx - xdy = O is not exact, but when multiplied by -, it becomes y dx - x dy = O y2 which is exact, and has for its solution x - C. y When multiplied by -, the above equation becomes xy dx dy O x y which is exact, and has for its solution log x - log y = c, which is transformable into the solution first found. Another factor that can be used with like effect on the same equation.1 X2 Any factor ~u, such as -, -, - used above, which changes y2 xy x an equation into an exact differential equation, is called an integrating factor. 15. The number of integrating factors is infinite. The number of integrating factors for an equation Mdx + Ndy = O, is infinite. For suppose ju is an integrating factor, then L (Mdx + Ndy)= du, and thus u = c is a solution. Multiplication of both sides by any function of u, say f(u), gives f (u) (Mdx + Ndy) = f(u) du; 22 DIFFERENTIAL EQUATIONS. [CH. II. but the second member of the last equation is an exact differential; therefore the first is also, and hence ff (u) is an integrating factor of the equation Mdx + Ncdy = 0; and as f(u) is an arbitrary function of u, the number of integrating factors is infinite. This fact is, however, of no special assistance in solving the equation. 16. Integrating factors found by inspection. Sometimes integrating factors can be seen at a glance, as in the example of Art. 14. Ex. 1. Solve y dx - x dy + log x dx = 0. Here logxdx is an exact differential, and a factor is needed for ydx - xdy. Obviously - is the factor to be ernployed, as it will not affect the third term injuriously, from the point of view of integration. The exact equation is then ydx-xdy log dx x2 X2, the solution of which reduces to x + y + log x + 1 = 0. Ex. 2. Solve (1 + xy)y dx + (1 - xy)x dy = O. Rearranging the terms, y dx + x dy + xy2dx - x2y dy = 0, that is, d(xy) + xAy dx - x2y dy - 0. For this, the factor I immediately suggests itself, and the equation x2y2 becomes d(xy) dx dyo x2y2 x y I x Integrating, -- + log = cl, 'xy + y and transforming, x = cyex. It will be well to try to find an integrating factor by inspection, before having recourse to the rules given in Arts. 17, 18, 19. ~ 16, 17.] RULES FOR INTEGRATING FACTORS. 23 Ex. 3. a(xdy + 2 y clx)= xydy. Ex. 4. (x3e - 2 my2) dx + 2 mAy dy = 0. Ex. 5. y(2 xy + ex) dx - exdy = O. 17. Rules for finding integrating factors. Rules I. and II. Rules for finding integrating factors in a few cases will now be given.* RULE I. When Mx + Ny is not equal to zero, and the equation is homogeneous, is an integrating factor of Mx + Ny Mdx + Nly = O. RULE II. When Mx - Ny is not equal to zero, and the equation has the form fi (xy) y dx + f2 (xy) x dy = 0, is an integrating factor. Mx - Ny PROOF: AMdx + Ndy = I { (MXl + INy) (dx + d + (MX _N) (c dy)}.illdx \+ y I \x = I y is an identity. This may be written, (a) Mdx + Ndy-1{ (Mx + Ny)d log xy + (Mx - Ny)d log x} Division of (a) by Mx + Ny gives Mdx + Ndy o Mx - Ny - x Mx + Ny = d*logxy + Mx d lo - Mx - Ny i hom Now if Mdx + Ndy is a homogeneous expression, x + Ny is homoX OMX + N'Y geneous and equal to a function of -, and Mcldx+Ndyd b x Mx + N =d logxy x+ f y d. logy,Y Mx + Ny y y or, since - = elog, For a discussion on and determination of integrating factors, see George Boole, Dfferential Equlations, pp. 55-90. 24 DIFFERENTIAL EQUATIONS. [CH. II. Mdx + Ndy x,F x lx + Ny = d. l~g xy -+ F log d log -, Mx+ NIVy 2 o 2 y which is an exact differential. On dividing (a) by Mx - Ny, it becomes, Mdx + Ndly 1 x+ N. ly x M-Tr —V =l 7 —1 d l ogxyoxy + d. log _, Mx - Ny Mx - Ny N1' and if Mdx + Ndy is of the form fi(xy)y dx + f2 (xy)xdy, tlis will be Mdx + Ndy f(xy)xy + ff(x)xy d log + d. og Mx - Ny 2 f(xy)xy - f(xy)xy lo = Fi(xy)d - logxy + d ci log-, -F (xy)d. logxy +, d log-, = F2(log xy)d - logxy + 2 d log -, y which is an exact differential. When M1x + Ny = 0, - =- Substitution for Min N X N AMdx + Ndy = O and integration gives the solution x = cy. When Mx- Ny =, =. Substitution for M in the differential N x N equation and integration gives the solution xy = c. Ex. 1. Solve (x2y - 2 xy2)dx - (x3 - 3 x2y)dy = 0. Ex. 2. Solve Ex. 3, Art. 9, by this method. Ex. 3. Solve y(xy + 2 x2y2)dx + x(xy - x2y2)dy = 0. 18. Rules III. and IV. dM cdN cli if ci Y RULE III. When dy - dx is a f nction of x alone, say f(), eN is an inte ti factor ef(x)dx is an integrating factor. ~ 18, 19.] RULES FOR INTEGRATING FACTORS. 25 For, multiplication of LMdx + Ndy = O by that factor gives, say, Mldx + Nldy = 0; and differentiation will show that dM,= dN. dy dx Ex. 1. (2 + y2 + 2 x) dx + 2ydy = 0. Ex. 2. (x2 + y2)dx- 2 xy dy- O. dN dM RULE IV. When d el is a function of y alone, say F(y), efF(y)d is an integrating factor. This can be shown in the same way as in the preceding rule. Ex. 3. Solve (3 2y4 + 2 xy)dx + (2 3y3 - 2)dy = O. Ex. 4. Solve (y4 + 2 y)dx + (xy3 + 2 y4 4 x)dy0. 19.* Rule V. x"m-l-ay.',l-, where K has any value, is an integrating factor of xayP (my dx + nx dy) = 0, for on using the factor, the equation becomes 1d(XKyKn) o. Moreover, when an equation can be put in the form x6yp (mny dx + nx dy) + xa1yPi (my dx + nx dy) = O, an integrating factor can be easily obtained. A factor that will lake xy(P (my dx + nx dy) an exact differential is x.. "1-ay.-' where K has any value; and a factor that will make xalîyP (rmnly (dx + n1x dy) an exact differential is xlml-1yKlnl-'-1 where K1 has any value. * See L'Abbé Moigno, Calcul Difféirentiel et Intér(ral (published 1844), t. II., No. 147, p. 355; Jolnson, DiTferential Equations, Art. 32. 26 DIFFERENTIAL EQUATIONS. [CH. II. These two factors are identical if Km - 1 - a = KlmI - 1 - a, and Kn - 1 - l = Klli - 1 - /1' Values of K and K1 can be found to satisfy these conditions. Ex. 1. Solve (3 - 2 yx2)cx + (2 xy2 - x3dy) = O. Rearranging in the form above, y2(y dx + 2 x dy) - x2(2 y dx + x dy) = O. For the first term a = 0, 3 = 2, m = 1, n = 2, and hence xKl-y2-1-2 is its integrating factor. Foi the second term a = 2, / = 0, m = 2, n - 1, and hence 2K' —1-2yK'-1 is its integrating factor. These factors are the same if K - 1 = 2 K - 1 - 2, and 2 K- 1 - 2 = K - 1. On solving for K and Kl, K = 2 = K', and therefore xy is the common integrating factor for both terms. The equation when made exact is xy{y2(y dx d- x(2 + 2 )- (2yd + xdy)} =0..2y4 X4y 2 2 -2 2 =c, or 2y2(y2 - 2)= c. Ex. 2. Solve (2x2y - 3 y4)Cd +(3 3 + 2 xy3) dy = O. Ex. 3. Solve (y2 + 2 x2y)dx + (2 x - xy)dy = 0. 20. Linear equations. A differential equation is said to be linear when the dependent variable and its derivatives appear only in the first degree. The form of the linear equation of the first order is a-,r PY =Q (1) where P and Q are functions of x or constants. The solution of + y = 0, that is, of - - Pdx y ~ 20.] LINEAR EQUATIONS. 27 is y = ce-JPx, or yex = c. On differentiation the latter forin gives efPdX (dy + Py dx) = O, which shows that eSPdx is an integrating factor of (1). Multiplication of (1) by that factor changes it into the exact equation, eJPdx (dy + Py dx) = efPdxQclx, which on integration gives yePdx -J'efPQdx - c, or y e-SPdx f PdxQc + c }. (2) The latter can be used as a formula for obtaining the value of y in a linear equation of the form (1).* The student is advised to make himself familiar with the linear equation and its solution, since it appears very frequently. Cly! Ex. 1. Solve x — ay = +1. clx dy This is linear since it is of the first degree in y and -- Putting it in cdx the regular form, it becomes dy a x +1 dx x x Here P a, and the integrating factor ePdx is 1 x Xa Using that factor, the equation changes to 1 dy x x +1 dx. xa Xa+-1 a+l ~*. Yx= x —~+ dx +c, xa 1 whence y = - 1+ a. 1-a a Gottfried Wilhelm Leibniz (1646-1716), who, it is generally admitted, invented the differential calculus independently of Newton, appears to have been the first who obtained the solution (2). 28 DIFFERENTIAL EQUATIONS. [Ci. II. The values of P and Q might have been substituted in the value of y as expressed in (2). Ex. 2. Solve dy + y dx Ex. 3. Solve cos2 x + ytan. dx Ex. 4. Solve (x + 1)dY - y ex(x + 1).+1 dx Ex. 5. Solve (x2 +1) + 2 xy 4 2. dx 21. Equations reducible to the linear form. Sometimes equations not linear can be reduced to the linear form. Il particular, this is the case with those of the form l + y Qyn (1) dx where P and Q are functions of x. For, on dividing by y'1 and multiplying by (- n - 1), this equation becomes (-+l)y-"y -n+Py-=(-n+1)Q; n + 1lyn dx+ (- n + 1) Py+l 1) Q; on putting v for y-"l', it reduces to +( -l n) Pv = (1 - n) Q, dx which is linear in v. Ex. 1. Solve y+- y= x2y6 dx x dy y-5 Division by y6 gives 6 -d + - 2 dx x dv 5 On putting v for y-5, this reduces to - - v - 5 x2, the linear form. C ~ IJ~VVIZlb dx x Its solution is v = y-5 = CX5 + 5x 2 * This is also called Bernoulli's equation, after James Bernoulli (1654 -1705), who studied it in 1695. ~ 21.] EXAM iPLES. 29 NOTE. In general, an equation of the form dy f'(y) x + f(Y)= Q, \ P and Q are functions of x, on the substitution of v for f(y) becomes dv + Pv dx which is linear. Ex. 2. Solve (1 + y2)clx = (tan- y - x)dy. This can be put in the form dx 1 tan-1 y _.+ xdy l+y2- 1 +y2' which is a linear equation, y being taken as the independent variable. Integration as in the last article gives the solution x = tan-1 y -1 + ce-tan-y. Ex. 3. Solve dy- 2y + -y 3 x2y. dx x Ex. 4. Solve dx+ xy = x dx 1 - x2 v dy Ex. 5. Solve 3 x(1 - x)y2- + (2x2 1) )y = ax3. EXAMPLES ON CHAPTER II. Solve the following equations: a dy ()2 dy i. (x + y)2 dY= c2. 5. (x2-yx2) -+ y2 + y2= 0. 2. y x_ _ 2. xy -y =/x + y2+. d y 22 ___ dclx x2 3. x, -y=x/z2+y2. dx,dy 2 xa 4. sec2xtanydx+sec2ytanxdy=0. 7 3 + 1- = y2 8. (2x-y+ l)dx +(2y-x- l)dy =0. dy y x + V1- x2 9 +( ) ( -2)2 d1 y y2 11. x - y. dx x 11. (X2 + Y2 - a2)X (lx + (X2 _ y2 _ b2)y dy = 0. 30 DIFFERENTIAL EQUATIONS. [CH. II. dy 4x 1dy 12. - (+ 14. x(1 - x2) +(2x2-)y = ax3. dx x+Y(X2 + 1)3 dx 13. x2y dx -(x3 + y)dyO = 0. 15. (a2 + y2 1)dx -2 xydy = 16. xdcx + ydy = m(xdy - ydx). 17. Integrate Ex. 16, after changing the variables by the transformation x = r cos, y = rsin0. 18. (1+eY)dx +ey 1 — dyO0. 21. d- - -xy. yy dx 19. d + y cos x sin 2 x. 22. yd + (a2y - 2 x)c = 0. 20. (x+ 1)- + 1-=2 e-y. 23. (1 6y2- 32y)d = 3xy2- 2. dx d24. y(x2 + y2 + a2)y + x(x+2 + 2 - a2) = 0. 25. (x2y3 + xy)dy = dx. 29. y dy + by2dx = a cos xdx. dy 26. y-= ax. 30. 2 xy x + (y2 - x2) dy = 0. 27. Va2 *xy + y = /a2 + 2- x. 31. (Xy2 _ )dx - xy dy O. dy dy -f- dy \~ 28. (x+ y) - (x - )=0. 32. y - x- b + 33. (3y+ 2x +4)dx-(4x +6 y +5)dy = 0..dy 34. (X3y3 + X2y2 + Xy + 1)y + (X3y3 - X2y2 - X + 1)x = 0. dx dy?z a 35. (2x2y2+y)dx-(x3y-3x)dyZ=0. 37. d+xxY= x. d3y y 3. (2 36. y2x +2 =xy-. 38. (x - y)2 = a2 dx clx dx ~ 22.] EQUATIONS NOT OF THE FIRST DEGREE. 31 CHAPTER III. EQUATIONS OF THE FIRST ORDER, BUT NOT OF THE FIRST DEGREE. 22. Equations that can be resolved into component equations of the first degree. In what follows, ~ will be denotecd by p. clx The type of the equation of the first order and nth degree is pn + pp^n-1 + P,23p-2 + * p-,_ + n- Pn ~ 0, (1) where P1, P2, *", P,, are functions of x and y. Two cases appear for consideration, viz.: (a) where the first member of (1) can be resolved into rational factors of the first degree; (b) where that member cannot be thus factored. In the first case (1) can take the form (p - Ri)(p - R2)... (p - (P- ) = 0. (2) Equation (1) is satisfied by a value of y that will make any factor of the first member of (2) equal to zero. Therefore, to obtain the solutions of (1), equate each of the factors in (2) to zero, and obtain the solutions of the n equations thus formed. The n solutions can be left distinct or combined into one. Suppose the solutions derived for (2) are fi (, y, cl) (, y ) = O(,, fY2 C 0,2) y, C = 0, where cl, c2,., c,, are the arbitrary constants of integration. These solutions are evidently just as general, if cl = c2= *. = cn, since all the c's can have any one of an infinite number of values; and the solutions will then be 32 DIFFERENTIAL EQUATIONS. [Ci. III. fi (, y, c) =-, f2 (x y, c) o,.,* (, (, c)= o. These can be combined into one equation; namely, fi (X, Y, c) 2 (I Y, C)c)... f (x,, c) = 0. (3) Ex. 1. p3 + 2 xp2 - y2p2 - 2 xy2p = 0, can be written p(p + 2 x)(p - y2)=. Its component equations are p=O, p 2 =0, p-/2=0, of which the solutions are y=c, y +x2=c, and xy + + 1 = 0, respectively. The combined solution is (y - c)(y + x2 - c)(xy + cy + 1)= O. When the equation in p is of the second order, sometimes the solution readily presents itself in the form (3) as ill the next exaniple. Ex. 2. Solve dy\ -ax3 = O. dy i a3 dx a 1 5 Integrating, y +- c = + 2 a2x2. Rationalizing, 25(y + c)2 = 4 ax5, or 25(y + c)2 - 4 a5 = 0. Ex. 3. Solve p3(x + 2 Y) + 3p2(x + y) + (y + 2 x)p = O. Ex. 4. Solve — ) = aC4. dlx Ex. 5. Solve 4y2p2 + 2pxy(3x + 1) + 3x3 = 0. Ex. 6. Solve p2 -7p + 12 = 0. 23. Equations that cannot be resolved into component equations. Methods which may be tried for solving equation (1) of the last article, when its first member cannot be resolved into rational linear factors, (case (b) Art. 22), will now be shown. That equation, which may be expressed in the form f(x,, p) = 0, may have one or more of the following properties. ~ 23, 24.] EQUATIONS SOLVABLE FOR y. 33 (a) It may be solvable for y. (b) It may be solvable for x. The case where it is solvable for p has been considered in the preceding section. (c) It either may not contain x, or it may not contain y. (d) It may be homogeneous in x and y. (e) It may be of the first degree in x anc y. 24. Equations solvable for y. When the condition (a) holds, f(x, y, p) = can be put in the form jy=F(x, p). Differentiation with respect to x gives /P dp\ which is an equation in two variables x and p; from this it may be possible to deduce a relation, (x, p, c) =.O The elimination of p between the latter and the original equation gives a relation involving x, y, and c, which is the solution required. When the elimination of p between these equations is not easily practicable, the values of x and y in terms of p as a parameter can be found, and these together will constitute the solution. Ex. 1. Solve x - yp =ap2. x - ap2 Here y = P Differentiating and clearing of fractions, (ap'2 + x) d = P(-p2) This can be put in tie linear form dx 1 ap dp p(l - D2)'x 1 -p2' D 34 DIFFERENTIAL EQUATIONS. [CH. III. Solving, x = P (c + a sin-1p). V/1 - p'2 Substituting in the value for y above, 1 y= -ap - (c + a sin-p). V/i +p2 Ex. 2. Solve y = x + a tan-lp. Ex. 3. Solve 4y -= 2 +p2. Ex.4. Solve xp2 - 2 yp + ax = O. 25. Equations solvable for x. When condition (b) holds, f(x, y, p) = O can be put in the form = F(y, p). Differentiation with respect to y gives 1 ( dp\ from which a relation between p and y may possibly be obtained, say, f(y,, c) -O. Between this and the given' equation p may be eliminated, or x and y expressed in terms of p as in the last article. Ex. 1. Solve x = y p2. Ex.2. Solve x = y + a logp. Ex. 3. Solve p2y + 2px = y. 26. Equations that do not contain x; that do not contain y. When the equation has the form f(y, p) = 0, and this is solvable for p, it will give dy dx = y)' which is integrable. ~25-27,] EQUATIONS HOMOGENEOUS IN x AND y. 35 If it is solvable for y, it will give Y= F(p), which is the case of Art. 24. When the equation is of the form f (, ) = 0, and this is solvable for p, it will give dy dx d= c (x), which is immediately integrable. If it is solvable for x, it will give x=F(p), which is the case of Art. 25. It is to be noticed that in equations having either of the properties (c) Art. 23 and not solvable for p, on solving for x or y the differentiation is made with respect to the absent variable. By differentiating in cases (a), (b), (c), there is a chance of obtaining a differential equation, by means of which another relation may be found between p and x or y in addition to the original relation. These two relations will then serve either for the elimination of p, or for the expression of x and y in terms of p. Ex. 1. Solve y = 2p + 3p2. Ex. 3. Solve x2 = a2(1 +p2). Ex. 2. Solve x(1 + p2)= 1. Ex. 4. Solve y2 = a2(1 + p2). 27. Equations homogeneous in x and y. When the equation is homogeneous in x and y, it can be put in the form F ( )=o. Fdx y= 36 DIFFERENTIAL EQUATIONS. [CH. III. It may be possible to solve this for d-, and then to proceed as in Art. 9; or to solve it for Y, and obtain X3 y = xf(p), which comes under case (a) Art. 23. Proceeding as in Art. 24, differentiate with respect to x, p = f(P) + xf (p) dx; dx f'(p) d)) whence -- f d x P-f(p) where the variables are separated. Ex. 1. Solve y2 + xyp - X2p2 = 0. Ex. 2. Solve y yp2 + 2px. 28. Equations of the first degree in x and y. Clairaut's equation. When the condition (e) Art. 23, holds, the equation, being solvable for x, and for y as well, comes unclear cases (a) and (b) considered in Arts. 24, 25. -However, tliere is one particular form of these equations of the first degree in x and y that is of special importance, namely, y = x + f(p), which is known as Clairaut's equations. Differentiation with respect to x gives p=-p+ + f' (p) - whence + f' (p) = O, dp. or C = 0. dx * Alexis Claude Clairaut (1713-1765), celebrated for his researches on the figure of the earth and on the motions of the moon, was the first who had the idea of aiding the integration of differential equations by differentiating them. He applied it to the equation that now bears his name, and published the method in 1734. ~28.] EQUATIONS OF FIRST DEGREE IN x AND y. 37 From the latter equation, it follows that p = c, and hence y= cx +f (c) is the solution. The equation x + f' (p) = O is considered in Art. 34. Any equation satisfying condition (e) can be put in the form y = fi (p) +f2(p). If f, (p) = p, it is in Clairaut's form. By proceeding as in Art. 24 and differentiating with respect to x there is obtained -f = (Ap) + fl' (p) + f2' (p) dx dx fC'z(p) + f2 (P) C' l p- f(p) -fi (p) which is linear in x; and from this a relation between x and p may be deduced. The student should be familiar enough with Clairaut's form to recognize it readily. Some equations are reducible to this forin; Ex. 2 is an illustration. Ex. 1. Solve y (1 + p)x + p2. dp Differentiating, p =1 + p + (x + 2p) _p clx dx ' x- = - 2p, dp which is linear. Solving, x =2(1 -p)+ ce-P; aind hence y = 2 - p2 + (1 + p)ce-P from the given equation. Ex. 2. Solve x2(y - px) = yp2 On putting x2 = u, and y2 = v, the equation becomes dv fdv'\2 ydu+ du which is Clairaut's form..'. V = Ceu + C2, and hence y2 = cX2 + C2. 38 DIFFERENTIAL EQUATIONS. [Cu. III. Ex. 3. Solve y = xp + sin-lp. Ex. 4. Solve e4 (p-1) + e2p2 = 0. Ex.5. Solve xy(y-px)= x +py. Solving for x or y may be of service in the case of equations of the first degree in p; this is illustrated in Ex. 6. Ex. 6. Solve y + 2xy = x +y2 dx The solution for y gives the equation y = x + vp, which is of the form discussed in Art. 24. The solution is y = c + e2. C - e2x 29. Summary. What has been said in this chapter concerning the equation f(x, y, p) = O, of degree higher than the first in p, may be thus suinued up: Either solve f(x, y, p) = 0 for p, and obtain a solution corresponding to each value of p; or, Solve for y or x, and, by differentiating with respect to x or y, obtain an equation, whence another relation between p and x or y can be found. This new relation, taken in connection with the original equation, will serve either for the elimination of p, or for the evaluation of x and y in terms of p; the eliminant or the values of x and y will be the solution. EXAMPLES ON CHAPTER III. x2 d 2 dy 1. z2(dY) 2 2 y2 2 2. y =p(x-b)+a 6. ayp2+(2x-b)p-y=O. 2. y =p(x - b) +_-. 7. y - px = x/1 + p2(x2 + y2). 3. xy2(p2 + 2) = 2py3 + X3.. Y- = +p2(X2y2 4. y -Xp + 42)=2. 8. (xp-y)2 —a(l+p2)(x2+y2)2. 4. y =- xp2+ x42-2. 5. p2-9p+18=0. - )2=p- +l. 10. 3p2y2 - 2 xyp + 4y2 x2 = 0. 11. (2 y2)(1 +p)2 2(x + y)(1 +p)(x +yp)+(x +yp)2 = O ~ 29.] EXAMPLES. 39 12. (py+nx)2=(y2+nx2)(1+p2). 16 (p a2 -1 p-) 13. y2(1-p2)-b. 14. (px -y)(py + x)= h2p. 17. x = a. 15. p2 + 2py cot x = y2. 18.? - 2px =f(xp2). 19. yp2 p(3 x2 - 2 y2) 6xy =. 20. p3- 4 xyp + 8y2 = 0. 21. p3 - (x2 + xy + y2)p2 + (x3y + x2y2 + xy3)p - X3y3 =. 22. p3 + mp2 = a(y + mx). 25. y(1 p2) b. 23. e3x(p - 1) + p3e2y = 0. 2 24 ( \ 2 ~ 26. y= px+-. 24. 1-y2+- p2-2 yp+ - 0.P3, ^ x ^x2- 27. y = 2px + y2p. 40 DIFFERENTIAL EQUATIONS. [CH. IV. CHAPTER IV. SINGULAR SOLUTIONS. 30. References to algebra and geometry. In this explanation of singular solutions, use will be made of a few definitions and principles of algebra and geometry; particularly of the discriminant in the one, and of envelopes in the other. Articles 31 and 32 will serve to recall some of them. The student is advised to consult a work on the theory of equations and a differential calculus concerning these points. 31. The discriminant. The discriminant of an equation involving a single variable is the simplest function of the coefficients in a rational integral form, whose vanishing is the condition that the equation have two equal roots. For example, the value of x in ax2 + bx + c = 0 is - V --; and so 2a the condition that the equation have equal roots is that b2 - 4 ac be equal to zero. The discriminant is b' - 4 ac; the equation b2 - 4ac = 0 will be called the discriminant relation. * Leibniz in 1694 (see footnote, p. 27), Brook Taylor (1685-1731), the discoverer of the theorern called by his naine, in 1715, and Clairaut (see footnote, p. 36) were the first to detect singular solutions of differential equations. Clairaut refers to these solutions in a paper published in the Memoirs of the Paris Acadeimy of Sciences in 1734. Their geometrical significance was first pointed out by Lagrange (see footnote, p. 155) in an article published in the iMemoirs of the Berlin Academy of Sciences in 1774, in which he also showed a way of obtaining them. The theory at present accepted is that expounded by Arthur Cayley (1821-1895) in an article in the Messenger of Mathemnatics, Vol. II., 1872. ~ 30-32.] THE ENVELOPE. 41 When the equation is quadratic, tle discrimiinant can be written iimmieciately; but when it is such that tle condition for equal roots is not easily perceived, the discriminant is found in the following way. The given equation being F= 0, form another equation by differentiating F with respect to the variable, and eliminate the variable between the two equations. For example, < (x, y, c) =0 may be looked on as an equation in c, its coefficients then being functions of x and y. The simplest rational function of x and y, whose vanishing expresses that the equation <p (x, y, c)= las equal roots for c, is called the c discriminant of cp, and is obtained by eliminating c between the equations, < (x,, c)=, d5 = 0. Thus the c discriminant relation represents the locus, for each point of which <b (x, y, c) = O has equal values of c. Similarly, the p discrinminant of f (x, y, p) = O, the differential equation corresponding to ) (x, y, c) = O, is obtained by eliminating p between the equations, 0If (,,?P)= O, -=0. clp Thus the p discriminant relation represents the locus, for each point of which f(x, y, p) = O has equal values of p. In order that there may be a c anc a p discriminant, the above equations must be of the second degree at least in c and p. In Art. 6 it was pointed out that these equations are of the same degree in c and p, and hence, if there is a p discriminant, there must be a c discriminant. 32. The envelope. If in <p (x, y, c) = 0, c be given all possible values, there is obtained a set of curves, infinite in number, of the same kind. Suppose that the c's are arranged in order of magnitude, the successive c's thus differing by infinitesinial amounts, and that all these curves are drawn. Curves corre 42 DIFFERENTIAL EQUATIONS. [CH. IV. sponding to two consecutive values of c are called consecutive curves, and their intersection is called an ultimate point of intersection. The locus of these ultimate points of intersection is the envelope of the system of curves. It is shown in works on the differential calculus, that the envelope is part of the locus of the equation obtained by eliminating c between (, y, c)= O, and d 0; dc that is, the envelope is part of the locus of the c discriminant relation. This might have been anticipated, because in the limit the c's for two consecutive curves become equal, and the c discriminant relation represents the locus of points for which > (x, y, c)-= will have equal values of c. It is also shown in the differential calculus, that at any point on the envelope, the latter is touched by some curve of the system; that is, that the envelope and some one of the curves have the same value of p at the point. 33. The singular solution. Suppose that f(x, y, p) 0 (1) is the differential equation, which has > (x,, c)=O (2) for its solution. It has been seen, in Arts. 4-6, that the system of curves which is the locus of f(x, y, p) =O is the set of curves obtained by giving c all possible values in (2). The x, y, p, at each point on the envelope of the system of curves which is the locus of (2), being identical with the x, y, p, of some point on one of these curves, satisfy (1). Therefore the equation of the envelope is also a solution of that differential equation. This is called the singular solution. It is distinguished from a particular solution, in that it is not contained ~ 33.] THE SINGULAR SOLUTION. 43 in the general solution; that is, it is not derived by giving the constant in the general solution a particular value. The singular solution may be obtained from the differential equation directly, without any knowledge of the general solution. For, at the points of ultimate intersection of consecutive curves, the p's for the intersecting curves become equal, and thus the locus of the points where the p's have equal roots will include the envelope; that is, the p discriminant relation df (1) contains the equation of the envelope of the system of curves represented by (2). In the next article, it will be shown that the p and c discriminant relations may sometimes represent other loci besides the envelope: that is, they may contain other equations besides the singular solution. The part of these relations that satisfies the differential equation is the singular solution. Ex. 1. y=xc~ + 1+ cl' which is in Clairaut's forin, has for its solution y = cx + av/1 + c2. This, on rationalization, becomes c2(a2 - x2) + 2 cxy + a2 - y2 = 0, and hence the condition for equal roots is x2 + y2 = a2. This relation satisfies the given equation, and hence is the singular solution. In this example, the general integral represents the system of lines y = cx + a /1 + c2, all of which touch the circle x2 + y2 = a2. Ex. 2. Find the general and the singular solutions of p2 + xp - y = O. Ex. 3. Find the general and the singular solutions of dy-/x = dxvAy. Ex. 4. Find the singular solution of x2p2 - 3 xyp + 2 y2 + x3 = 0. Ex. 5. Find the general and the singular solutions of (1+ dy)- +) (1 - 44 DIFFERENTIAL EQUATIONS. [CiI. IV. 34. Clairaut's equation. In finding the solution of Clairaut's forni in Art. 28, there appeared the equation x +f'(p)= 0, (3) which is as important as the equation dp= 0, that appeared with it. The foregoing shows what part equation (3) plays in solving Clairaut's equation. On differentiating y px + f(p) with respect to p, (3) is obtained. The elimination of p between these two equations gives the p discriminant relation, which here represents the envelope of the system of lines y =cx +f(c) represented by the general solution. 35. Relations, not solutions, that may appear in the p and c discriminant relations. It has been pointed out that the p discriminant relation of f(x, y, p)= 0 represents the locus, for each point of which f(x, y, p) -- 0 will have equal values of p; and that the c discriminant relation of qb (x, y, c) = ), the general solution of the former equation, represents the locus for each point of which p (x, y, c) = O will have equal values of c. It is known also that each point on the envelope of the systeni, (x, y, c) -.0 is a point of ultimate intersection of a pair of consecutive curves of that system; and, moreover, that at each point on the envelope there will be two equal values of p, one for each of the consecutive curves intersecting at the point; and that, therefore, the singular solution, representing the envelope, must appear in both the 2p and the c discriminant relations. But the question then arises, may there not be other loci besides the envelope, whose points will make f(x, y, p) = give equal values of p, or Imakle (x, y, c) = give equal values of c? In other words, while the p and tle c discriminant relations must both contain the singular solution, which represents the envelope if there be one, may they not each contain something else? ~ 34-37.] TAC-LOCTUS AND NODAL LOCUS. 45 36. Equation of the tac-locus. At a point satisfying the p discriminant relation there are two equal values of p; these equal p's, however, may belong to two curves of the system that are not consecutive, but which happen to touch at the point in question. Such a point of contact of two non-consecutive curves is on a locus called the tac-locus of the system of curves. The equations representing the tac-locus, while thus appearing in the p discriminant relation, will not be contained in that of the c discriminant; since the touching curves, being non-consecutive, will have different c's. Ex. Examine y2(1 + p2) = r2. -/r2 _ y2 Solving for p, p = - - Y2 Integrating and rationalizing, y2 + (z + c)2= r2. The general solution, therefore, represents a system of circles having a radius equal to r and their centres on the x axis. The c discriminant relation is y2 - r2 = 0, and that of the p discriminant is y2(y2 - r2) = 0. Thus the locus of the latter is made up of the loci y = + r and of y = O counted twice. The equations y = + r, that appear in both the p and the c discriminant relations, satisfy the differential equation, and hence form the singular solution; they represent the envelope. The equation y = 0, as is apparent on substitution, does not satisfy the differential equation. Through every point on the locus y = 0, two circles of the system can be drawn touching each other; that equation, therefore, represents the tac-locus. The student is advised to make a figure, showing the set of circles, their. envelope, and the tac-locus, as it will help him to understand this and the preceding articles. *37. Equation of the nodal locus. The c discriminant relation, like that of the p discriminant, may contain an equation having a locus, the x, y, p, of whose points will not satisfy the differential equation. 46 DIFFERENTIAL EQUATIONS. [CH. IV. The general solution (b (x, y, c) = O may represent a set of curves each of which has a double point. Changing the c changes the position of the curve, but not its character. These L FIG. 1. curves being supposed drawn, the double points will lie on a curve which is called the nodl locus. In the limit two consecutive curves of the system will have their nodes in coincidence upon the nodal locus. The node is thus one of the ultimate points of intersection of consecutive curves; and, therefore, the equation of this locus must appear in the c discriminant relation. But in Fig. 1, where A, B, *.., are the curves and L is the nodal locus, at any point the p for the nodal locus L is different from the p's of the particular curve that passes through the point; and hence the x, y, p, belonging to L at the point, will not satisfy the differential equation. A B L FIG. 2. And, in general, the x, y, p, at points on the nodal locus will not satisfy the differential equation; for the case would be exceptional where the p at any point on the nodal locus would ~ 38.] CUSPIDAL LOCUS. 47 coincide with a p for a curve of the general solution passing through that point; where, in other words, the nodal locus would also be an envelope, as in Fig. 2, in which A, B,.., L, have the same signification as in Fig. 1. Ex. xp2 - (x - a)2 = O has for its general solution 3 1 y + c = - 2 axr; that is, 9(y +c)2=x(x - 3a)2. The p discriminant relation is x(x - a)2 = 0, and that of the c discriminant, x(x - 3 a)2 = 0. The relation x = O satisfies the differential equation; hence it is the singular solution and represents the envelopelocus. x = a x=3 a The relation x - a = 0, which appears Y only in the p discriminant, does not satisfy / the differential equation; it represents the \ tac-locus. And x - 3a = 0, which is in the c discriminant, does not satisfy the original equation; it represents the nodal locus. Figure 3 shows some of the curves of the system, the envelope, the tac, and the nodal loci. 38. Equation of the cuspidal locus. The general solution ( (x, y, c) = may represent a set of curves each -_ ix of which has a cusp. These curves being supposed drawn, the cusps will lie on a curve called the cuspidal FIG. 3. locus. It is evident that in the limit two consecutive curves of the system will have their cusps coincident upon the cuspidal locus, the cusps thus being among the ultimate points of intersection; and hence the cuspidal locus will appear in the locus of the c discriminant relation. Moreover, the p's at the cusps of consecutive curves will evidently be equal; and therefore the cuspidal locus will appear 48 DIFFERENTIAL EQUATIONS. [CH. IV. in the locus of the p discriminant relation. Like the nodal locus, it will not, in general, be the envelope. Ex. 1. The differential equation p2 + 2p- y =0 (1) has for its general solution (2 x3 + 3 xy + c)2 - 4(2 + y)3 = 0. (2) The p discriminant relation is x2 + y = 0, (3) and the c discriminant relation is (X2 + y)3 = 0. Equation (1) is not satisfied by (3), and hence there is no singular solution; x2 + y = 0 is a cusp locus. Ex. 2. The equation 8 ap = 27 y has for its general solution ay2 = (x - c)3; the p discriminant relation is y = 0, and the c discriminant relation is y4 = 0. FG. 4. The equation y = O satisfies the differential equation, and therefore is the singular solution. It is also the equation of the cusp locus. Figure 4 illustrates this example. This is one of the very exceptional cases where the cusp locus coincides with the envelope. 39. Summary. When the loci discussed above exist, then in the p discriminant relation will appear the equations of the envelope locus, of the cuspical locus, and of the tac-locus; and in the c discriminant equation will appear the equations of the envelope locus, of the cuspical locus, and of the nodal locus.* * See Edwards, Diffprential CalculuIS, Arts. 364-366; Johnson, Differential Equations, Arts. 45-54; Forsyth, Dff'erential Equations, Arts. 23-30; an article by Cayley, " On the theory of the singular solutions of differential equations of the first order" (Messelner of Mathematics, Vol. II. [1872], pp. 6-12); an article by J. W. L. Glaisher, "Examples illustrative of Cayley's theory of singular solutions" (Messenger of MVathematics, Vol. XII. [1882], pp. 1-14). ~ 39.] EXAM3PLES. 49 The p discriminant relation contains the equations of the envelope, cuspidal and tac loci, once, once, and twice respectively; and the c discriminant relation contains the equations of the envelope, cuspidal and nodal loci, once, three times, and twice respectively.* EXAMPLES ON CHAPTER IV. Solve and find the singular solutions of the following equations: 1. xp2-2yp+ax =0. 3. y2 - 2x+2(X2 - 1)=m2. 2. xp2 +- x2yp + a3 = 0. 4. y = xp + V/b2 + a2p2. 5. y= xp -p2 6. Examine Exs. 2, 4, 20, 26, Chap. III., for singular solutions. 7. Solve 4p2 -9 x, and examine for singular solution. 8. Investigate for singular solution 4 (x - 1)(x - 2)p2 - (3x2 - 6x + 2)2 =0. 9. Solve and examine for singular solution (8p3 - 27)x = 12p2y. 10. p2(2 - a2)- 2pxy + 22 - b2 = 0. 11. (px -- y) (x - py)= 2 p. * This is proved in an article by M. J. M. Hill, " On the c and p discriminant of ordinary integrable differential equations of the first order" (Proc. Lond. Mathi. Soc., Vol. XIX. [1888], pp. 561-589). This article supplements Cayley's, mentioned above. For further information see Professor Chrystal: "p discriminant of differential equations of first order" (Nature, Vol. LIV. [1896], p. 191). E 50 DIFFERENTIAL EQUATIONS. [CH. V. CHAPTER V. APPLICATIONS TO GEOMETRY, MECHANICS, AND PHYSICS. 40. The student will remember that, after deducing the methods of solving various kinds of algebraic equations and working through lists of these equations, he made practical applications of the knowledge and skill thus acquired, in the solution of problems. In the process of finding the solution of one of these problerns, there were three steps: first, forming the equations that expressed the relations existing between the quantities considered in the problem; second, solving these equations; and third, interpreting the algebraic solution. In the case of differential equations, the same procedure will be followed. The three preceding chapters have shown methods of solving differential equations of the first order. This chapter will be concerned with practical problems, the solution of which will require the use of these methods. The problems will be chosen for the most pait from geometry and mechanics; and it is presupposed that the student possesses as much knowledge of these subjects as can be acquired from elementary text-books on the differential calculus and mechanics. As in the case of algebraic problems, there are three steps in obtaining the solution of the problems now to be considered: First, forming the differential equations that express the relations existing between the variables involved. Second, finding the solution of these equations. Third, interpreting this solution. ~ 40-42.] APPLICATIONS. 51 There will be only two variables involved in each of these problems, and hence but a single equation will be required. The choice of examples for this chapter is restricted, because differential equations of the first order only have so far been treated. 41. Geometrical problems. The student should review the articles in the differential calculus that deal with curves; in particular, those articles that treat of the tangent and normal, their directions, lengths, and projections, and the articles that discuss curvature and the radius of curvature. This review will be of great service in helping him to express the data of the problem in the form of an equation, and to interpret the solution of this equation. The character of the geometrical problems and the method of their solution will in general be as follows. A curve will be described by some property belonging to it, and from this its equation will have to be deduced. This is like what is done in analytic geoinetry, but here the statement of the property will take the form of a differential equation; the solution of this differential equation will be the required equation of the curve. 42. Geometrical data. The following list of some of the principal geometrical deductions of the differential calculus is given for reference. It will be of service in forming the differential equations which express the conditions stated in the problems, or, in other words, give the properties belonging to the curves whose equations are required. Suppose that the equation of a curve, rectangular co-ordinates being chosen, is y =f(x), or F (, y)= O, and that (x, y) is any point on this curve. Then dy is the idx slope of the tangent at the point (x, y), i.e. the tangent of the dx angle that the tangent line there makes with the x-axis; dx dy is the slope of the normal; the equation of the tangent at (x, y), 52 DIFFERENTIAL EQUATIONS. [CH. V. X, Y, being the current co-ordinates, is Y- y = cy (X - ); and dx the equation of the normal is Y- y - (y x) the intercept of the tangent on the axis of x is x- y d; the intercept dy dy of the tangent on the axis of y is y - x d the length of the cdx tangent, that is, the part of the tangent between the point and the x-axis, is y V +()2; the length of the normal is y 1 +(-); the length of the subtangent is y -; the d exn dy length of the subnormal is y cd; the differential of the length d_____ x of the arc is -1 +Û(y dy, or +1 dy dx; the differential of the area is y dx or x dy. Again, let the equation of the curve in polar co-ordinates be f(r 0)= 0, orr= F(), and (r, 0) be any point on the curve. Then the tangent of the angle between the radius vector and the part of the tangent to the curve at (r, 0) drawn back towards the initial line, is dO r -; if 0 is the vectorial angle, / the angle between the radius dr vector and the tangent at (r, 0), and 4 the angle that this tangent makes with the initial line, -r = + 0; the length of the polar subtangent is r2 d-; the length of the polar subdr normal is -; the differential of the length of the arc is 1 + 2 2c/ dr, or +r2d+; if p lenote the length of the perpendicular froin the pole upon the tangent,* then - Williamson, Differential Calculus, Art. 183; Edwards, Differential Calculus for Beginners, Art. 95. ~ 43.] GEOMETRICAL EXAMPLES. 53 i 1 1 cfd\2 12 22 + }f4fitl that is, - 2+ d where = - p2 = O.u r7 43. Examples. Ex. 1. Determine the curve whose subtangent is n times the abscissa of the point of contact; and find the particular curve which passes through the point (2, 3). Let (x, y) be any point upon the curve. The subtangent is y - Theredy fore, the condition that must be satisfied at any point of the required curve, in other words, the given property of the curve, is expressed by the equation dx. dy Integration gives n log y = log cx, whence, yn = Cx. This represents a family of curves, each of which passes through the 3n origin. For the particular curve that passes through (2, 3), c must be 2, and the equation is 2 y =3n x. When n =1, the required curve is any one of the straight lnes which pass through the origin; the equation of the particular line through (2, 3) is 2y = 3x. When n = 2, the curves having the given property are the parabolas whose vertices are at the origin, and whose axes coincide with the x-axis; the particular parabola through (2, 3) has the equation 2y2 = 9x. When n =, the required curve is any one of the system of semicubical parabolas that have their vertices at the origin and their axes coinciding with the axis of x. What curves have the given property when n = ~? When n = 2? Ex. 2. Find the curve in which the perpendicular upon the tangent from the foot of the ordinate of the point of contact is constant and equal to a; and determine the constant of integration in such a inanner that the curve shall cut the axis of y at right angles. 54 DIFFERENTIAL EQUATIONS. [CH. V. Let (x, y) be any point on the curve. The equation of the tangent at (x, y) is Y- 4-)dy, Y _y dy (X- x); the length of the perpendicular from (x, 0), the foot of the ordinate, -y upon the tangent is /1 +(dy Therefore, the given property of the curve is expressed by the equation (1) ____- - a; fromthis, (2) ady = dx (1) (d> )2 <xY2 -a2 ^ \dx) integration gives * cosh-1 -Y + c; a a whence (3) Y =cosh + c. a \a ) It is aiso required that there be found the particular one of these curves that cuts the y-axis at right angles. This means that for this curve, dy= - when x = 0. Now differentiation of (3) gives dx 1 dy 1. x a dx a \a therefore c = 0; and hence y x - = cosh x a a' the equation of the catenary. Ex. 3. Determine the curve in which the subtangent is n times the subnormal. Ex. 4. Determine the curve in which the length of the arc measured from a fixed point A to any point P is proportional to the square root ofthe abscissa of P. Ex. 5. Find the curve in which the polar subnormal is proportional to the sine of the vectorial angle. Ex. 6. Find the curve in which the polar subtangent is proportional to the length of the radius vector. See McMahon, Hyperbolic Functions (Merriman and Woodward, Higher Mathematics, Chap. IV.), Arts. 14, 15, 26, 39; Edwards, Integral Calculus for Beginners, Arts 28-44. ~ 44, 45.] TRAJECTORIES. 55 44. Problems relating to trajectories. An important group of geometrical problems is that which deals with trajectories. A trajectory of a family of curves is a line that cuts all the members of the family according to a given law; for example, the line which cuts all the curves of the family at points equidistant from the x-axis, the distance being measured along the curves of the family. Another example of a trajectory is the line that cuts the curves of the family at a constant angle. When the angle is a right angle, the trajectories are called orthogonal trajectories; when it is other than a right angle, the trajectories are said to be oblique. Only these two classes of trajectories will here be discussed. 45. Trajectories, rectangular co-ordinates. Suppose that f (x y, a)= (1) is the equation of the given system of curves, a being the arbitrary parameter; and that a is the angle at which the trajectories are to cut the given curves. The elimination of a from (1) gives an equation of the form q. y, )= O, (2) the differential equation of the family of curves. Now through any point (x, y) there pass a curve of the given system and one of the trajectories, cutting each other at an angle a. If m is the slope of the tangent to the trajectory at this point, then dy + tan a, dy, - -tan a dx dd cly By definition m is d for the trajectory; hence the differential equation of the system of trajectories is obtained by substituting this value of m for - in (2); this gives dx 56 DIFFERENTIAL EQUATIONS. [CH. V. dy dx - +- tan a for the differential equation of the system of trajectories; and the solution of this is the integral equation. If a is a right angle, dx m -- i dy and hence the differential equation of the system of orthogonal trajectories is obtained by substituting -d for d- in (2); this dy dx gives ~xy,- d o. (5) Integration will give the equation in the ordinary form. 46. Orthogonal trajectories, polar co-ordinates. Suppose that f(r, O, c)= 0 (1) is the polar equation of the given curve, and that (ro, d)= (2) is the corresponding differential equation, obtained by eliminating the arbitrary constant c. The tangent of the angle between the radius vector and the tangent to a curve of the given sysci tem at any point (r, 0) is r —. If m is the tangent of the angle between this radius vector and the tangent to the trajectory through that point, 1 dr r dO since the tangents of the curve and its trajectory are at right angles to each other. Hence the differential equation of the ~ 46, 47.] TRAJECTORIES. 57 required trajectory is obtained by substituting - - d_ for r-,, dû dr dr dr' or, what cornes to the same thing, - r- - for in (2); this cid d0 gives Ço (, - "2 T)= (3) as the differential equation of the required system of trajectories. 47. Examples. Ex. 1. Find the equation of the curve which cuts at a constant angle whose tangent is î' al the circles touching. a given straight line at a given point. Take the given point for the origin, the given line for the y-axis, and the perpendicular to it at the point for the x-axis. The given system of circles then consists of the circles which pass through the origin and have their centres on the x-axis; its equation is y2+ — 2 ax O, (1) a being the variable parameter. The elimination of a gives the differential equation of the system of circles; namely, dy y2 -x2 --. (2) dx 2 xy The differéntial equation of the system of trajectories is obtained by dy substituting for l in equation (2) the expression clx dy m dlx n 1 mdy' n dx and this gives on reduction (nx2 - ny2 + 2 mxy)dx + (imy2 - mx2 + 2 nxy)dy =- 0. (3) The integration of this homogeneous equation gives x2 + y2 =- 2c(my + nrx), (4) c being the constant of integration; this represents another system of circles. The trajectory is orthogonal if n = 0; equation (4) then becomes x2 + y2 = 2 cny, 58 DIFFERENTIAL EQUATIONS. [CH. V. which represents the orthogonal system of circles; these circles pass through the origin and have their centres on the y-axis. Ex. 2. Find the orthogonal trajectories of the system of curves r"n sin nO = an. Differentiation eliminates the parameter a, and gives dr1 + r cot nO = 0, dO the differential equation of the system. The differential equation of the system of trajectories is obtained by substituting - r2 d( for d tlis gives dr d0 r2d + r cot no = 0; dr separating the variables, integrating, and simplifying, rL cos nO = c, c being an arbitrary constant; this is the equation of the system of orthogonal curves. Ex. 3. Find the orthogonal trajectories of a series of parabolas whose equation is y2 = 4 ax. Ex. 4. Find the orthogonal trajectories of the series of hypocycloids 2 2 2 x+3 y -= a. Ex. 5. Find the equation of the system of orthogonal trajectories of a series of confocal and coaxial parabolas r = 2 — 1 + cos 0 Ex. 6. Find the orthogonal trajectories of the series of curves. r = a + sin 5 0. Ex. 7. Given the set of lines y = cx, c being arbitrary, find all the curves that cut these lines at a constant angle 0. 48. Mechanical and physical problems. The student should read in some text-book on mechanics the articles in which the elementary principles anc formulae relating to force and motion are enunciated and deduced. The truth of the following definitions and formulae will be apparent to one who understands the first principles of the calculus and the principles of me ~ 48.] MECJHANICAL AND PHYSICAL EXAMPLES. 59 chanics as set forth in elementary works that do not employ the calculus. If s denotes the length of the path described by a particle moving for any period of time; t, the tinie of motion, usually estimated in seconds; and v, the velocity of the moving particle at any particular point or instant; then will ds dt v, and dv d= the acceleration of the moving particle at any point dt of its path. Ex. 1. A body falls from rest; assuming that the resistance of the air is proportional to the-square of the velocity, find (a) its velocity at any instant; (b) the distance through which it has fallen. In this case the equation for the acceleration is dv dv = g - KV2, or, putting n- for ~, dt g g = 2 2 dt whence gdv = dt. g2 _ n2v2 lnv n h d Integrating, tanh- -— = nt + c; whence, = tanh(nt + c). g g But c = 0, since v = 0 when t =0. Therefore v ds tanh nt; dt n whence, on integration, s + c = log cosh nt. But s=0whent=0,.. c=O; therefore = g log cosh nt. n" 60 DIFFERlENTIAL EQUATIONS. [Cri: V. Ex. 2. Find the distance passed over in time t by a particle whose acceleration is constant, determining the constants of integration so that at the time t = 0, vo is the velocity and so the distance of the particle from the point frorn which distance is measured. Ex. 3. The velocity possessed by a body after falling vertically from rest through a distance s is found to be /2 gs. Find tlie height through which it lias fallen in terms of the time. EXAMPLES ON CHAPTER V. 1. Determine the curve in which the length of the subnormal is proportional to the square of the ordinate. 2. Determine the curve in which the part of the tangent interceptedby the axes is constant. 3. Determine the curve in which the length of the subnormnal is proportional to the square of the abscissa. 4. Find the equation of the curve for which a differential of the arc is K times the differential of the angle rnade by its tangent with the x-axis, multiplied by the cosine of this angle; and deternine the constant of integration so that the curve touches the x-axis at the point from whicl the arc is measured. 5.. Find the equation of the curve where the length of the perpendicular from the pole upon the tangent is constant and equal to -. 6. Find the equation of the system of curves that make an angle whose tangent is - with the series of parallel lines x cos a + y sin a =p, p being the variable parameter. 7. Find the orthogonal trajectories of the systein of parabolas y-= ax2. 8. Find the orthogonal trajectories of the system of circles toucling a given straight line at a given point. 9. Find the orthogonal trajectories of 2- +-2- = 1, where X is arbitrary. 10. Find the orthogonal trajectories of the series of hyperbolas xy = K2. 11. Determine the orthogonal trajectories of the system of curves r" = a" cos n; therefrom find the orthogonal trajectories of the series of lemniscata r2 - a2 cos 2 0. 12. Find the orthogonal trajectories of - ) cos 0 - a, a being the parameter. ~48.] iEX AMPLES. 61 13. Find the orthogonal trajectories of the series of logarithmic spirals r = a0, where a varies. 14. Determine the curve whose tangent cuts off from the co-ordinate axes intercepts whose sum is constant. 15. The perpendiculars froin the origin upon the tangents of a curve are of constant length a. Find the equation of the curve. 16. Find the equation of the curve in which the perpendicular froin the origin upon the tangent is equal to tile abscissa of the point of contact. 17. Find the equation of a curve such that the projection of its ordinate upon the normal is equal to the abscissa. 18. Find the equation of the curve in whicl, if any point 1' be taken, the perpendicular let fall from the foot of its ordinate upoii its radius vector shall cut the y-axis where the latter is cut by the tangent to the curve at P. 19. Find the curve in which the angle between the radius vector and the tangent is n times the vectorial angle. What is the curve when n = 1? When 1n? 20. I)etermine the curve in which the normal makes equal angles with tle rLadius vector and the initial line. 21. Finld the curve the length of whose arc ineasured from a given point is a mean proportional between the ordinate and twice the abscissa. 22. Find the equation of the curve in which the pepeendicular from the pole upon the tangent at any point is k times the radius vector of the point. 23. If = 2( e2 - - 1 ), 1 )find the equation of the curve, r being p2 I (1 -l c) \ r / the radios vector of aniy point of the curve, and pj the perpendicular from the pole upon the tangent at that point. 24. Find the orthogonal trajectories of the cardioids r a(1 - cos 0). 25. Show tliat tie system of confocal and coaxial parabolas y = 4a(x + a) is self-orthogonal. 26. Show that a system of confocal comics is self-orthogonal. 27. Find the curve such that the rectangle under the perpendiculars from two fixed points on the normals be constant. 28. Find the curve in wlich the product of the perpendiculars drawn from two fixed points to any tangent is constant. 62 DIFFERENTIAL EQUATIONS. [CH. V. 29. The product of two ordinates drawn froin two fixed points on the x-axis to the tangent of a curve is constant and equal to K2. Find the equation of the curve. 30. Determine the curve in which the area enclosed between the tangent and the co-ordinate axes is equal to a2. 31. Find a curve such that the area included between a tangent, the x-axis, and two perpendiculars upon the tangent from two fixed points on the x-axis is constant and equal to K2. 32. The parabola y2 = 4 ax rolls upon a straight line. Determine the curve traced by the focus. 33. Determine the curve in which s = ax2. 34. The equation of electromotive forces for an electric circuit containing resistance and self-induction is E = i+ Lidt where E is the electromotive force given to the circuit, R the resistance, and L the coefficient of induction. Find the current i: (a) when E = f(t); (b) when E= O; (c) when E= a constant; (d) when E is a simple harmonic function of the time, E,, sin wt, where E,, is the maximum value of the impressed electromotive force, and w is 27r times the frequency of alternation; (e) when E = E1 sin wt + E2 sin (bwt + 0). 35. The equation of electromotive forces in terns of the current i, for an electric circuit having a resistance R, and having in series with that resistance a condenser of capacity C, is E = Ri + ci, which reduces on differentiation to the form di i 1 dE dt RC P dt E being the electromotive force. Find the current i: (a) when E =f(t); (b) when E = O; (c) when E = a constant; (d) when E = E, sin wt. 36. Given that the equation of electromotive forces in the circuit of the last example, in terms of the charge q, is E: R dq q E=R +d<ï+( - dt C find q: (a) when E =f(t); (b) when E = 0O; (c) when E = a constant; (d) when E = E, sin wt. 37. The acceleration of a moving particle being proportional to the cube of the velocity and negative, find the distance passed over in time t, the initial velocity being vo, and the distance being measured froin the position of the particle at the time t = O. ~ 49.] LINEAR EQUATIONS. 63 CHAPTER VI. LINEAR EQUATIONS WITH CONSTANT COEFFICIENTS. 49. Linear equations defined. The complementary function, the particular integral, the complete integral. Equations of an order higher than the first have now to be considered. This chapter and the next will deal with a single class of these equations; namely, linear differential equations. In these, the dependent variable and its derivatives appear only in the first degree and are not multiplied together, their coefficients all being constants or functions of x. The general form of the equation is dny + ld"-y _ pd"t-2y dx' — + P~d-~-l+ Pa2 d1- 2 + + ny = X, (1) where X and the coefficients Pl, P, "', P,, are constants or functions of x. If the derivative of highest order, -, has a clx,, coefficient other than unity, thle members of the equation can be divided by this coefficient, and then the equation will be in the form (1). The linear equation of the first order has been treated in Art. 20. It will first be shown that the complete solution of (1) contains, as part of itself, the complete solution of dny + P - +... + P" = 0. If y /=y be an integral of (2), then, as will be seen on substitution in (2), y = cy,, c, being an arbitrary constant, is also an integral; similarly if y =2 y y3, y=, *' = y,, be integrals of (2), then y = c2y2, '., y_ cy,, where c2,..., c, are arbi 64 DIFFERENTIAL EQUATIONS. [Ci. VI. trary constants, are all integrals. Moreover, substitution will show that y CiYi + C2y2 +. CnyL (3) is an integral. If y,, Y2, *'., y,, are linearly independent,* (3) is the complete integral of (2), since it contains n arbitrary constants and (2) is of order n. If y = u be a solution of (1), then y = Y+ u, (4) where Y-= cyl + c2I- + + Cn"Yl, is also a solution of (1); for the u -' '-<*t-i of Yfor y in the first member of (1) gives zero, and that of u for y, by hypothesis, gives X. As the solution (4) contains n arbitrary constants,. it is the complete solution of equation (1). The part Y is called the conplemectacy fiOnctiotn; and the part u is called the CparticClar integr'Cl.''j The general or complete solution is the sum of the complementary function and the particular integral. 50. The linear equation with constant coefficients and second member zero. The equation d -- + p dY + P..+ ', (1) where the coefficients P,, P2,.*, P,,, are constants, will first be treated. On the substitution of ex for y, the first member of this equation becomes (mn + Pn'1-l + ***+ -P,) elx; and this will be equal to zero if n + P,' + Pi1 + + Pn= 0. (2) * See Note F for the criterion of the linear independence of the integrals Yi, y2,., YV,. t This use of the term particular integral is to be distinguished from that indicated in Art. 4. t The method of solving the linear differential equation with constant coefficients, shown in this article, is due to Leonhard Euler (1707-1783), one of the most distinguished mathematicians of the eighteenth century. ~ 50, 51.] LINEAR EQUATIONS. 65 This nay be called the acxilicary equation. Therefore, if rz have a value, say m,, that satisfies (2), y = emlx is an integral of (1); and if the n roots of (2) be mnl, m2,... mn the complete solution of (1) is y = cieml + c,2e2 +.. + c,~enx. (3) Ex. 1. Solve dy 3 - 54= 0. dIX2 (lx Here equation (2) is qn2 + 3 m - 54 0; solving for m, m = 6, - 9. Hence the general solution of the equation is y cle6x + c2e-9. Ex. 2. If d - m2y = O, isow that dx' y = cle? + c2e-mx = A cosh mxq + B sinh rnx.* Ex. 3. Solve 2 d+5 -12x =0. dt2 cdt Ex. 4. Solve 9 2 + 1 16 x = 0. c*2 cdz 51. Case of the auxiliary equation having equal roots. When two roots of (2) Art. 50 are equcl, say mn, ancd m,, solution (3) becomes y = (CI + -2) emlx + cCem3x... + cemnx. But, since c, + c2 is equivalent to only a single constant, this solution will have (n -- 1) arbitrary constants; and hence is not the general solution. In order to obtain the complete solution in this case, suppose that nl =n i, + h; the terms of the solution corresponding to m,, m,, will tlen be y = ceml + C2e(ml+7)x which can be written y = em7n (cI + cex). * See McMahon,?yperbolic Functions (IM(rriman and Woodward, Higher Mathematics, Chap. IV.), Arts. 14 (Prob. 30), 17, 39. F 66 DIFFERENTIAL EQUATIONS. [CH. VI. On expanding ehx by the exponential series, this becomes y = em [c l + c2 (1 h + +...)] = em"x [c + + hx ( + hxc + 7z + +.)] - e"ml (A + Bx + C2h2x2 + terms proceeding in ascending powers of h), where A -= c + Iantd B = ch. Now let h approach 0, aind solution (3) Art. 2 takes the forin y = emix (A + Bx) + c3em3x +.. + cne.nx. As h approaches zero, cl and c, can be taken in such a way that A and B will be finite. 4 bSh 94t. ~ k. / f If the auxiliary equation 'have three roots equal to rn1, by similar reasoning it can be shown that the corresponding solution is y = emlx(ci + c2x + c3x2); and, if it have r equal roots, that the corresponding solution is y = emlX(cl + C2x + ***+ Crr-1).* The form of the solution in the case of repeated roots of the auxiliary equation is deduced in another way in Art. 55. yd3 d2y Ex. 1. Solve 5 d 4y 0. dx3 dx2 d4y d8y 9d2y 11dy Ex. 2. Solve d-d -9 -11 - -4y =0. 52. Case of the auxiliary equation having imaginary roots. When equation (2) Art. 50 has a pair of imaginary roots, say ml = a + if/, m2= a- if3 (i being used to denote /-1), the corresponding part of the solution can be put in a real form simpler, and hence more useful, than the exponential form of Art. 50. * See George Boole, Differ ential Equations, Chap. IX., Art. 7. The separate integrals, emix, xemîx, x2em1x,..., are analogous to the equal roots of an algebraic equation. ~ 52, 53.] THE SYMBOL D. 67 For, cle(a+iP)+ c2e(a-iP) = eaz(ceipz + c2e-iP) = ecX 1 C (cos 3x + i sin px) + c2 (cos 3x - i sin /]x) = ea (A cos px + B sin fx) = (cosh ccx + sinh ax) (A cos /x + B sin 3x). If a pair of imaginary roots occurs twice, the corresponding solution is y = (Ci + c2x) e( +i)x + (c3 + c4x) e(a-i), which reduces to y = ex (A + Alx) cos Px + (B + Bx) sin p/. d2y dy Ex. 1. Solve + 8 - + 25y = 0. dx2 dx The auxiliary equation is m2 + 8 m + 25 = 0, the roots of which are m = - 4 + 3 i; and the solution is y = e-4x(cl cos 3 x + c2 sin 3 x). Ex. 2. If d4y -n4y = 0, show that dX4 C cl cos mx + c2 sin mx + C3 cosh mx + C4 sinh x. CEY dx dx dx2'Y dx Ex. 3. Solve d-4 Y + 8 d -8 d + 4Y = 0. 53. The symbol D. By using the symbol D for the differential operator -, equation (1) Art. 50 can be written dx (D" + PAD-1 + ~ + Pn) y = 0, (1) or, briefly, f(D) y = 0. (2) The symbolic coefficient of y in (1) is the same function of D that the first member of equation (2) Art. 50 is of m; and, therefore, the roots of the latter equation being mn, m2,, mn,, equation (1) may be written (D - mi)(D - m2).. (D - mD)y = 0. (3) Hence the integral of (1) can be found by putting its symbolic coefficient equal to zero, and solving for D as if it were.an ordinary algebraic quantity, without any regard to its use as an operator; and then proceeding as in Art. 50 after the roots of equation (2) of that article had been found. Moreover, it is thus apparent that the complete solution of (1) or (3) is made up of the solutions of 68 DIFFERENTIAL EQUATIONS. [CIH. VI. (D- ) y = 0, (D- )y= 0,., (D - m,) y = 0. This symbol D will be of great service. 54. Theorem concerning D. One of the theorems relating to D is, that when the coefficient of y in (1) Art. 53 is factored as if D were an ordinary algebraic quantity, then the original differential equation will be obtained when D is given its operative character, no matter in what order the factors are taken. Thus, an equation of the second order -( _ ~ + +) dy y = o, when expressed in the symbolic form, is 2o- -(a +/)D + /iY = 0; this on factoring beconies (D - a) (D -- f/) y 0. Replacing D by -, the latter equation becomes dx (dx ) dx Operating on y with d _, this becomes dx,dx \dx and, operating on the second factor with the first, d2y + (a~, -( )~.+ 0. If the factors had been written ini tle reverse order, (D - )(D - a) y =, and expanded as above, the same result would have been obtained. It is easily shown that the theorem holds for an equation of the third anc any higher order. It will be noted that the symbolic factors, when used as operators, are taken in order from right to left. Other theorems ~ 54, 55.] REPEATED ROOTS. 69 relating to D will be proven when a reference to then happens to be required.* 55. Another way of finding the solution when the auxiliary equation has repeated roots. The form of the solution when the auxiliary equation (2) Art. 50 has repeated roots can be found in another way; namely, by employing the symbol D. According to Art. 53, the solutions corresponding to the two equal roots nm, of this equation are the solutions of (D - m)2y = 0. On writing this in the form (D - m,) (D - ml) y = O and putting v for (D - m,) y, the above equation becomes (D - m) v = 0, the solution of which is v = ce"il. Replacing v by its value (D - nm) y, (D - i) y = clelx, which is the linear equation of the first order considered in Art. 20; its solution is y = e"'x (cl + cx). Similarly the solutions corresponding to three equal roots mi are the solutions of (D - m)3y = 0, which may be written (D - m) (D - m)2 y = 0. On putting v for (D - m,)2y, solving for v, and replacing the value of v as before there is obtained (D - tnl)" y = cenx. Putting o for (D - mi) y and proceeding as before, (D - mn) y = emx (cl + c2), * See Forsyth, Differential Equations, Arts. 31-35, for fuller information concerning the properties of D. 70 DIFFERENTIAL EQUATIONS. [CH. VI. the solution of which is y = emlx (ex2 + c2x + c3), where c = c. 2 It is obvious that if m1 is repeated r times, the corresponding integrals are y = emlx (C1 + C2x + * + cr-1). 56. The linear equation with constant coefficients and second member a function of x. In this article will be considered the equation dY, + Pl + -. + Pa Y = X, (1) the first member of which is the same as that of equation (1) Art. 50, and the second member a function of x. It was pointed out in Art. 49 that the complete integral of (1) consists of two parts, -a complementary function and a particular integral, the complenientary function being the complete solution of the equation formed by putting the first member of (1) equal to zero. The problem now is to devise a method for obtaining the particular integral. In the symbolic notation, (1) becomes f(D) y= X (2) and the particular integral is written y = X. f(D) 57. The symbolic function. It is necessary to define 1 wf(D) (D-X, which, as yet, is a mere symbol without meaning. f (D) 1 For this purpose it may be said: X is that function of x f(D) which, when operated upon by f(D), gives X. The operator 1 (D) according to this definition, is the inverse of the f (D) operator f(D). It can be shown from this definition and Art. 54, that -- can be broken up into factors which may be taken in any order, or into partial fractions. ~ 56, 57.] THE PARTICULAR INTEGRAL. 71 For example, the particular integral of the equation d2y cdy 1' (U- +/)a- +U-f is X; D2 - (a + /) D + x; and this can be put in the form i X. (D - a) (D- ) Now apply (D - a) (D - ) to this, arranging the factor of the latter operator conveniently, as is allowable by Art. 54; this gives (D - /) (D - ) (D (D X; and since D - a, acting upon * -- X must by definiD- aD - - tion give X, this becomes D - * -- X, which is D-/3 DX by the definition of f(D) This reduction shows that the particular integral might equally well have been written 1 X. (D-/3)(D-a) Also, D —X + may be written in the form D2 - (a + 3)D + af3 1' 1 1 /X a-[ D-a D —, ' which is obtained by resolving the operator into partial fractions. The result of operating upon this with D2- (a + ) D + is 1 _ (D-/)(D- ) X- (D - )(D -) XP a — D -a - or 1 (D- )X -(D- a) X; and finally, X. and finally, X. 72 DIFFEIRENTIAL EQUATIONS. [CCI. VI. The statement immediately preceding this example can easily be verified for the general case by a method similar to that used in this particular instance. 58. Methods of finding the particular integral. It is thus apparent that the particular integral of equation (2) Art. 56, namely, — X, may be obtailnec in the two following ways: (a) The operator may be factored; then the particular integral will be f() 1 1 1 X D —m D —ms D —m~, D - m1 D - 2 D - m,, On operating with the first symbolic factor, beginning at the right, there is obtained 1. 1... 1 emn e-mnx Xdx; D - ml D - m2 D - m_,,then, on operating with the second and remaining factors in succession, taking then from right to left, 'there is finally obtained the value of the particular integral, namely, emixje(m2-m1..... J e-mnx X(dx)n. (b) The operator may be decomposed into its partial fractions f + _... + D - mi1 D - DDn and then the particular integral will have the form Neml fe-mlxXclx + N em2xje m-2Xdx-+-... + Nen fe-"&nXdx. Of these two methods, the latter is generally to be preferred. Since the methods (a) and (b) consist altogether of operations of the kind effected by --- upon X, the result of the latter - C- 1 operation should be remembered. Now, -- X is the particular integral of the linear equation of the first order, ~ 58, 59) SHORT METHODS. 73 ldy d_ ay = X, which has been discussed in Art. 20; its value is ea e-ax Xdx. The term ceax in the solution of this equation is the complementary function. Ex. 1. Solve (dy 5d + 6y = e4x. dx2 dx This equation, written in symbolic form is (D2 - 5 D t- 6)y = e4, or (D -3) (D -2) = e4; hence the complementary function is y = cie3x + c2e22; and the particular integral is y -.1 e4~_= I - 1 )e4 D-3 D-2 \D-3 D-2 2 —2 = e3e-3xe4xX - e2e-2xe4xdx = e4x e4 e4x ad( hence the general solution is y = Cle3x + 2e2x + e4. Ex. 2. Solve d-y=2+5x. dx2 Ex. 3. Solve d -2 d + y = 2 e2x. dx" dx Ex. 4. Solve 3 d2Y-8 + 12=X (dx3 dx2 C(x 59. Short methods of finding the particular integrals in certain cases. The terms of the particular integral which correspond to terms of certain special forms that may appear in the second member of the equation, can be obtained by methods that are much shorter than the general methods shown in the last article. The special forms occurring in the second member which will be discussed here are: 74 DIFFERENTIAL EQUATIONS. [CH. VI. (a) eax, where a is any constant; (b) xm, where m is a positive integer; (c) sin ax, cos ax; (d) ea V, where V is any function of x; (e) xV, where V is any function of x. 60. Integral corresponding to a term of form eax in the second member. The integral corresponding to eax in the second member of the equation f(D)y = x, is f- eax; this will be shown to be equal to ef-. 1(a) Successive differentiation gives D"eax = cteax; the terms appearing in f(D) are terms of the form D, n being an integer; therefore f(D) ea= f(a)ea. Operating on both mernbers with 1 f(D)' 1 1 f(D1D) cax _ f(a) e~z; and this, since - ald f(D) are inverse operators and f(a) f(D) is only an algebraic multiplier, reduces to eax= f(a) _-1( eax whence eax ea. f(D) f(a) The method fails if a is a root of f(D) = 0, for then - eax = o eax. f(D) In this case, the procedure is as follows. Since a is a root of f(D)= O, (D - a) is a factor of f(D). Suppose that f(D) (D - a) q (D); then ____ (x__' _ 1 1 a_' xeax 1eaX _ 1 ea 1 1 a f(D) (D - ) q (D) (D - a) (a) c (a) ~ 60, 61.] TERM 0F FORM x"', 75 If a is a double root of f(D)= 0, then D - c enters twice as a factor into f(D). Suppose that f(D) (D - a)2 (D); then 1 i 1 11 e~ _ 2ea OaXzzz eax f (D) (D )2 (D) (D _- a)2 j (a) 2 ) (a) The method of procedure is obvious for the case when a is a root of f (D)= 0, r times. Ex. 1. Solve + y z 3 + c-x + 5 e2 CX2 Written in symbolic form, this equation becomes (D3 + 1)y- 3 + e-x + 5 e2x Here the roots of f(D)= are -1, -; hience the complementary function is ce- + e 2c cosxC 3+ c2 sin The particular integral is 1 (3 eO' + e-x + 5 ex);,O'4 +1 substitution of O and 2 for D, on account of the first and third terms, gives 3 + e2. But -1 is a root of D3 + 1, hence, factoring te denominator, 1 1 1 e-r e-xz z-" - D3+ 1 D + D2 -D + 1 D 1 3 on substituting - 1 for D in the second factor; the last expression is equal to -; hence the complete solution is x3 3 X3 y = ce-x + e (ci cos2 sin 2) + 3 4+ 5 2x + xe-" O 3 Ex. 2. Find the particular integrals of Exs. 1, 3, Art. 58, by the short method. Ex. 3. Solve - =(e + 1)2. Ex. 4. Solve d 2 d + y =3 e'. 61. Integral corresponding to a term of form xm in the second member, mn being a positive integer. Wheu -- x" is to be f(D) 76 DIFFERENTIA L EQ UTIONS. [Ci. VI. evaluated, raise f(D) to the (- l)th power, arranging the terms in ascending powers of D; with the several terms of the expression thus obtainecl, operate on x"1; the result will be the particular integral corresponding to x"n. It is obvious that terms of the expansion beyond the rtth power of D need not be written, since the result of their operation on x"1 woulc be zero. Ex. 1. Solve (D3 + 3 D2 + 2 D)y = x2 The roots of f(D) are 0, - 2, -1; and hence the complementary function is cl + c2e-2 +- c3ex-. The particular integral 1 _2_ 1 (+- _ -.. -- I 2 2 D+ 3D2+ 3 2- 2 (1 - - D+ D2 +...)x2 2D21 = - z3X+ (2x2-9x+21), 1 D - x being merely sx lx. The complete solution is y -- + ce-2 - cae- + - (2 2 - 9x + 21). The operator on x2 could equally well have been put in the former ad thi gies te reIt alre ti. Oe it tik tt it voul and this gives thé result already obtaiiied. One iniglit think tliat it would be necessary to add another tc rn, D', in tilis foirm of the operator; but the result for tlis term would be a tiumerical constant; and this is already included in the compleinentary function. Ex. 2. Solve Ex. 2, Art. 58, by this method. diy Ex. 3. Solve + 8 y = X4 + 2 X + 1. coxe 62. Integral corresponding to a term of form sin cx or cos ax in the second member. Successive differentiation of sin ax gives D sin ax = a cos ax, D2 sin ax - a2 sin ax, ~ 62.] TERM 0F FORM iSTN ax OR COS ax. 77 D)3 sill ax = - co as ctXo D4 sin ax = - a4 sin ax =- (- ca)2 sin cax and, in general, (D2)" sin ax = (- a)'" sin ax. Therefore, if q (D2) be a rational integral function of D2, <( (D2) sin ax =< (- a2) sin ax. From the latter equation and the definition in Art. 57, it follows, since < (- a) is merely an algebraic multiplier, that sin ax sin ax. ~ (D2) ( (- c) Similarly, it can be shown that i i --- cosC ax = - cos ax; (2) COS ^(- 2)C and, more generally, tlat - sin (ax + a)= sin (a + a), ~ (D') 2(-.) _ i1 and - -7^cos(ax+ oa)=, - cos(ax + a). P(D~) ~p(- a2) Ex. 1. Solve + d d- cos 2; dx' dX2 dxs O that is, (D3 + D2 D -- l)y _ cos 2 x. The complementary function is clex + e-x(c2 + c3x); and the particular 1 1 1 integral = - D —1 cosI x- cos 2 Dga+ D2 - D - 1 - D+I D2 —1 =D-1 cos2x (D -— 1)os2x cos 2 x 1 cos 2 x (D2 - 1)2 25 2 cos 2 x _ 2 sin 2 x -c2 25 25 hence the complete solution is y = clez + e- (C2 + c3 x) sin 2 x cos 2 25 25 The number, - 4, might have been substituted for D2 at any step in the work. 78 DIFFERENTIAL EQUATIONS. [Ci. VI. d2y Ex. 2. Solve -i + a2y -- cos ax. The complementary function is ci cos ax + c2 sin ax; the particular integral is -- cos ax = 1 cos ax; and thus the method fails. " D2 + at2 - a2 + a2 In this case, change a to a + h; this gives for the value of the particular integral, - cos(a + x)h; this expression, on the application of the D2 + a2 principle above and the expansion of the operand by Taylor's series, i h-a2 becomes --- (cos ax - sin ax hx - cos ax ~ +..). -(a + h)2 +2 12. The first term is already contained in the complemnentary function, and hence need not be regarded here; the particular integral will accordingly be written 1_(_in hx2 (x sin ax + cos ax + terms with higher powers of h); on making h approach zero, this reduces to x s1 ax. 2 a The complete integral is y - c cos ax + c2 sin ax + xsn ax2a Ex. 3. Solve c d y - 4 y 2 sin x. dx2 Ex. 4. Solve dy +y = sin3 - cos2 - x. dx3 63. Integral corresponding to a term of form eaxV in the second member, V being any function of x. Since De"a V= eaxD V + aea" V= e<a(D + a) V; and D2ea V- aeax(D + c) V + e-ZD (D + ac) V= eax(D + a) V; and, in general, as is apparent from successive differentiation, D'e-LV = eax(D + 6)' V; therefore, f (D)eax V= eaf(D + a) V. (1) Now put f(D + ct)V=;i; then V= — I+ I. A1so,9 V f(D + a) will be any function of x, since V is any function of x. Substitution of this value of Vin (1) gives f (D)e f(D. VD - e V1; ~ 63, 64.] TERM OF FORMi xV. 79 whence, operating on both members of this equation with (D)' and transposing, f 1 1 e^n:V, _ eJ7, 1__I1 f (D) f (D + a) where VT, as las been observed, is any function of x. Ex. 1. Solve d- +y = xe2. The complete solution is y - c1 cos x + C2 sin x + X-1 e2Z D)2 +1 By the formula just obtained, x e - — ' ~2x; 1xe^ = e2 -1-X -= e2,1 ---x: D2 + 1 (D + 2)2 - 1 5 + 4 D+ D2 '2x and this, by the methocl of Art. 61, gives the integral (5x - 4). ' 25 The complete solution is p2x y = c1 cos x + C2 sin x + 25 (5-4). d 2y dy Ex. 2. Solve î + 3 + 2y - e2 sin x. d2y Ex. 3. Solve - + 2 y -= e3x + ex cos 2 x. 64. Integral corresponding to a term of the form xV in the second member, V being any function of x. Suppose that a terml of tle form xT occurs in f (D)y = X. Differentiation shows that DxV =-xDV + V, D2. V = D2 V -+ 2 DV, D"zx V= xD V + nDl1 V;, or, as it may be written, = xD)V+ Y-D )V. Therefore, f(D)x V = xf (D) V + f'(D) V (1) 80 DIFFERENTIAL EQUATIONS. [CII. VI. As ill Art. 63, the samLe formula holds for the operator f(D) 1 and the inverse operator - 1(D) To show this, put f/(D)) V=; then V =f- ( TV; and V being any function of x, FT will also denote any function of x. The substitution of this value of V in (1) gives 1 1 f(D)x V xJ,- = +f VI+.f '(D) V I1 Operating on both members of this equation with — ), anc transposing, 1 f (D)) 1 (D) ' - V f(D) f(D) t D)j where ) clednotes 6(D I(D) The particular integral corresponding to expressions of the form xrV, where r is a positive integer, can evidently be obtained by successive applications of this methodd* Ex. 1. Fiinc the particular integral of Ex. 1, Art. 63, by this method. 1 1 2 D e2x The particular integral = - xe2X = x e2x _e2 D2 + 1 D2 + I (D2 + 1)Y e2:e _ —(5x- 4). 25 Ex. 2. Solve - 1+ 4 y = xsinx. Ex. 3. Solve d2 y x 2 cosx. EXAMPLES ON CHAPTER VI. d4y 2. (D5 -13D +26 D2 + 82D + 104)y =0. * See Johnson, Differential Equations, Art. 120; Forsyth, Differential Equations, Art. 46. ~ 64.] EXAMPLES. 81 d&y d'2y dy e2x d2y 3. d+2 dY+dY = e2x + x. 12. a2y = sec ax. dx2 dx2 4 d 4. (-2~ + 4 y = sin 3 x ex. 13. d-2 d -y =. dj2y d + 14. (D2 + n2)y = eSX4. 95. 5~ d+y-y-x+ e = x. 15. i- a4y = x4. 6. (D2 - Ca2)y = eax + en + 5. d32y d2y dy dy d3y d2Yc 7. -3 -6+2 8y =x. 16. (2+ = 2dd= x. dx3 dîC2 dIx c-XX3 dx 8d _y d __ dYay 8. + 2= c x2(c + bx). 17. d4 y- ecx osx. àI34 àx - dx2 dX d:~y dy d~'y dy 9. 1 + 12 y =. 18. d- + y sin 2 x. dclxD dx d2+ ' 23. iB - 7 -a -i /. 4_y dy Y7-y - y 2 y 10. 222 dx2 -~+ îj4 = COS mnx. 19. d dy e2(1 + x). il. y22y + y - X-2cos. 20. dY - 2 dy +4y cosx. L IX4 dx2 - dx2 dx d1~-y d(y C1dy 21. + 3 2' +3 + y - e-x. 22. (DI - 2D3 -D2 + 4D4)y = x2e. 23. day e dy 23. 3 +3 Y- = xex s + ex. d~x3 dix2 dx 24. - y - xsinx +(1 d- x2)e". dx 2 25. (D2 - 4 D + 3)y e cos 2x- + cos 3 x. 26. (D-3D' D2+ 4 D -- 2)y = e + cosx. 27. (D2 -9D+20)y = 20x. 28. (D3 - 3 D -+ 4)yC = e3. 29d?. -x/3 Cdx xX, 2 30. Show that ND +- M X, where N and M1 are constants, is equal (D - o)2 + i2 to twice the real part of 2 ( ) X operated on by ND + M; 2 ip/ D - a ipf that isi to ea in px Sea cos -x. Xdx -e" cos _ )e-^ sm sin px. Xdx. (Jolihnson, Difj. Eq., Art. 106.) G 82 DIFFERENTIAL EQUATIONS. [CiC. VI1. CHAPTER VII. LINEAR EQUATIONS WITH VARIABLE COEFFICIENTS. 65. The homogeneous linear equation. First method of solution. This chapter will treat of linear equations ini which the coefficients are functions of x. For the most part, however, it will discuss only a very special class of these equations, namely, the homogeneous linear equation. A homogeneous linear equation is an equation of the form d"y dn-:y dyx" n ydx n d + +p~x-lad _ +***+ pn-l xdY+ PnY=x, O) where p,, p2 *.., p,, are constants, and X is a function of x. This equation can be transformed into an equation with constant coefficients by changing the independent variable x to z, the relation between x and z being z = log x, that is, x = ez. If this change be made, then dy dydz 1 dy dx dz dx x dz d2y _1 d2y dyi dx2 x- \dz2 dz)' d3y i (d3y d9y dy\ dx^ ~ tdXn 12 dzn-l dz dx" x Xndx n 12d-1' + ~ 65.] HOMOGENEOUS LINEAR EQUATIONS. 83 On putting D for -, and clearing of fractions, these equations become dy x =Dy, dx x 2d2= D (D -1) y, 3 Y = D (D-1)(D- 2)y, (2) dny, Y= D (D - 1)(D -2)... (D -n + 1) y; and, in general, the operators only and not the operand being indicated, xr = D (D - )... (D-r + ). (3) dxr Hence, the substitution of ez for x in (1) changes it into the form D (D-1)... (D-n+l)+pD(D 1)... (D-n+2)+. +p.y =z, (4) where Z is the function of z into which X is changed. Equation (4), having constant coefficients, can be solve by the methods of the last chapter. If its solution be y f(z), then the solution of (1) is y = f(log x). Therefore equation (1) can be solved by putting x equal to ez, thus changing the independent variable from x to z, which reduces the given equation to one with constant coefficients; and then solving-'the newly formed equation by the methods of the preceding chapter. Ex. 1. Solve x2 d - + y = 2 ogx. On changing the independent variable by putting x equal to eZ, this equation becomes {D(D - 1)- D + l}y = 2z. 84 DIFFERENTIAL EQUATIONS. [Ci. VII. On solving this equation by the methods of Arts. 51, 61, the complete integral is found to be y =ez(c + c2z) + 2 z + 4, which, in terms of x, is y - x(c + c2 log x) + 2 log x + 4. Ex. 2. Solve x2d2 + y 32. 66. Second method of solution: (A) To find the complementary function. The solution of X"d t" + pl +-l... + PtY = X (1) can be found clirectly, without explicitly making the transformation shown in Art. 65. The first member of (1) becomes, when x"' is substituted for y, mr(m (- 1)(n- 2)... (m - n + 1)4- (p1 - 1) *. * (m - n + 2) +*... + n }m. Therefore, if m (m - 1) *. (, m - n + 1) + Ii (m - 1). (2 - n + 2) + ** +p- = O, (2) the substitution of e' for y makes the first member of (1) vanish; and then x"' is a part of the complementary function of the solution of (1). Therefore, if the n roots of (2) are n, m2,..., m,W the complementary function of the solution of (1) is C1Xmi + C22+... + *** + Cn the c's being arbitrary constants. It will be noticecl that the first member of (2) is the same function of m as the coefficient of y in equation (4) Art. 65 is of D. Therefore, corresponding to an integral y = x11 of (1), there is an integral y = e"lZ of (4-) Art. 65; and hence, as has ~ 66, 67.] SECOND METIOD OF SOLUTION. 85 already been seen, an integral of (1) can be obtained by substituting log x for z in the integral of (4) Art. 65. Therefore, if (2) has a root mn repeated r times, the corresponding integral of equation (4) Art. 65 being y = emlz(ci + c2z + c3z2 + *.. + CrZr-), the integral of (1) is y -= x c, + C, log + *. + c(log X)r-. Similarly, if (2) have a pair of imaginary roots a ~ i/3, the corresponding integral of (4) Art. 65 being y = eaz(cl cos fz + c2 sin /z), the integral of (1) is y = xa C1 cos (p/ log x) + C2 sin P (log ) }. 3dy d2'y dy Ex. 1. Solve X3 d + 3 d2 d+ y = 0. dx3 dX2 îOx Substitution of x'" for y gives (m3 + 1)xm = 0; the roots of this equation are - 1, 14 +73 i 2 and hence the solution is /- x { C2 COS 2gX) + C3 sin log ). Ex. 2. Find the complementary functions of Exs. 1, 2, Art. 65, by this method. Ex. 3. Solve d 4Y+ 63d3 + 9X2d+ 3x d y = 0. jaX- dX3 CIX2 dx 67. Second method of solution: (B) To find the particular integral. In this article and the two following, the particular integral of edny dn-ly n + plxn-l +... +P = (1) xwill be fond. will be found. 86 DIFFERENTIAL EQUATIONS. [CH. VII. dz Since the symbol D in (3) Art. 65 stands for d-z that is, for x, this equation may be written dx d d ( d _A / d,l x-d == Txl- ~1 " n+; dxr dx dx dxc therefore (1), when 0 is substituted for x- therein,t takes the dx form 0(0-1)... (0-n- +l)+P0(0-1)... (- n-+2)+.. P Y-+-X. The coefficient of y here is the same function of 0 as the first member of (2) Art. 66 is of n. Let this equation be written in the symbolic form f(0)y:= X. (3) The method deduced. in Art. 66 for finding the complementary function of (3) may on making use of the symbol 0 be thus indicated: If the n roots of f(0) = O are 01, 02,..., 0,,, the complementary function of (3) is C1:x0 + c2x02 +... + Cn,,n, the c's being arbitrary constants. The particular integral of (3), after the manner used in the case of f(D) y = X, in the last chapter, may be expressed in the form — X. A method for evaluating X must now f(O) f(0) be devised. 68. The symbolic functions f(O) and. As to the direct symbolf(0), it is to be observed that its factors are commutative. For example, * See Forsyth, Differential Equations, Arts. 36, 37. t Thus 0 stands for the - of Art. 65, which was there symbolized by dz D; but as D had already been used to indicate d, this new symbol is required. ~ 68, 69.] THE SYMBOL C FUNCTION f(O). 87 (0 - a)(0.- 8)y = X2 - ( + - 1) + ay; d3 d2Y and (0- a)(0 - - )(0- y)y=3 d + (3 - a - - y)2y dx3(3c-p7;)xdx2 + (UP + y7 + y - - - + 1) - d yy = o; and this shows, by the symmetry of the constant coefficients, that the order of the operative factors is indifferent. The student can complete the proof of this theorem concerning f(0) for himself. If X be defined as that function of x which when f(0) operated upon by f(0) will give X, then it can be slown, by the method followed in Art. 57 in the case of the symbolic function, that - can be decomposed into factors which are f(D)' f(0) commutative; and also that it can be broken up into partial fractions. 69. Methods of finding the particular integral. It is thus apparent that the particular integral of (3) Art. 67 can be found in the following ways: (a) The operator - may be expressed in factorial form, and f- X will then become f(0) X; 0- a 6 -a2 O - an here the operations indicated by the factors are to be taken in succession, beginning with the first on the right; the final result will be the particular integral. (b) The operator -- may be broken up into partial fracf(o) tions, and consequently X be thus expressed, (a 2 -Yi + c,. -0 -a 88 DIFFERENTIAL EQUATIONS. [Cii. VII. the suni of thle results of the operations indicated will be the particular integral. Since the methods (a) and (b) are made up of operations of the kind effected by - upon X, the result of the latter 9 - a operation should be impressed upon the memory. Now, 0- X is the particular integral of the linear equation of the first order x d _ay X. dx The particular integral of this equation, by Art. 20, is found to be xafx x 'Xdxl; therefore 1 X = xa Jx-a-Xdx. 0 —g Hence the inethod (c) will give xai a2-al- j.. *** nan- 1-1 X (dx. ) as the value of the particular integral of (3) Art. 67; and the method (b) will give Nxalf-al-lXdx + XN2a2-a2-Xdx... + N,,xan -an-'lXdx as its value. When a term ( is one of the partial fractions of (O - (Y i (b), the operator n- must be applied to X, n times in succession; and this will give the result xc p -1 y x-f.,.f-a-1 X(dx)n. Ex. 1. Solve x2 d 2 x -4 y= x4. dx2 -lx Here f(0) is 0(0 - 1) - 2 0 - 4,which reduces to (0 - 4)(0 + 1). ~ 70.] TERM OF FORM Xa. 89 Hence the complementary function is clx4 + -, and the particular 1 x integral is ( 4) x4. (0 - 4)( + 1) On using partial fractions, the latter will be written 1- I X4- _ 1 x'; this reduces to 5\0-4 0+1 1 1 X4 X-4 l+4dx - I — ~1 I1+4dx, which on integration is x4 log x x4 5 25 The complete integral is, therefore, c2 X4 log X y= Ci4 +- the term - is included in the term c1x4 of the complementary function 25 Ex. 2. Solve x2 d 5x + 4y = 4. 02 y dy d+X2 dx 70. Integral corresponding to a term of form xa in the second member. In the case of the homogeneous linear equation, as in that of the equation discussed in the last chapter, methods shorter than the general one can be deduced for finding the particular integrals which correspond to terms of special form in the second member. For instance, 1 _ f(0) f(m) PnooF: ( - " = m', x-) xm = m'2xm, and for any positive integer r, (x xl) 7n = r xm. Now f() is a rational integral function of 0, and therefore f (0)x = f(m)xm. Applying 1 to both members of this equation, and transposing, f(m) being merely an algebraic multiplier, 1 x = 1 fm. f(O) f(m) 90 DIFFERENTIAL EQUATIONS. [CH. VII. If m is a root of f(O)= 0, then f(m) = O; and hence thie method fails. II this case f(O) cai be factored into (0 - mn)((0); the particular integral then becornes 1 x%, which reduces to 1 * - xm; this is 0 - n() ( 0(n) 6-rn equal to xmlogx. If m is repeated as a root r times, f(O) = F(O) ( - n)r, and the corresponding integral is -1 1 x-', which has tle value x-(log x)r F(mn) (O - n)' r! F(m) Ex.1. Solve x2 ~+7x +5y = x5 Here f(0) is 02 + 6 + 5; hence the complementary function is clX-3 + c2x-2; and the particular I X>5 integral is x5, which, on substituting 5 for 6, becomes -xo The 02 + 60 + 5 60 complete solution is thus X5 y = C1X-3 + C2X-2 +Ex. 2. Find the particular integrals of Exs. 1, 2, 3, Art. 69 by this method. 71. Equations reducible to the homogeneous linear form. There are some equations that are easily reducible to the homogeneous linear form; anc hence also to the formn of the linear equation with constant coefficients; for, as has been seen in Art. 65, these two forms are transforluable into each other. Any equation of the form * dny dn-ly (a + bx)d- + P1 (ac + bx)"- d y +, dy + Pn- (a + bx) + Pny =F( ), ( 1) where the coefficients P1,..., Pn, are constants, is transformed into the homogeneous linear equation, Z, (dI+ Pl z,-l dn- PP y 2,- Cin-2y+ cldz b dz" —l b2 dz"-2 + nP dy P F (2) 6b-' dz ibl' ( b This is known as Legendre's equation. See footnote, p. 105. ~ 71.] EXAMPLES. 91 when the independent variable is changed from x to z, by the substitution of z for a + bx. If the solution of (2) be y= F(z), the solution of (1) is y-= F(a + bx). If et had been substituted for a + bx, the independent variable thus being change froin x to t, there would have been derived a linear equation with constant coefficients. Ex. 1. Solve (5 + 2 x)2 6(5+2x)y+8y =O. Ex. 2. Solve (2 - 1)33 + (2 X-l) -2 y =. d(X3 dxc EXAMPLES ON CHAPTER VII. 1 d3y 4d2y 5 dy 2y __ ___+ 1. C d3 X cdx z2 2dx X3 2. x2 + 4 x + 2y dx'2 c(x,di3 lydx2 dx 3. o — 3- 32 dzY + 7 xa _ 8 y = O. d'2y dy 4. (x + a)2 2- 4(x + a) + 6 y = x. 6. X3 cY+ 2X2 2+ 2yo=10(x+ ) 6(x y d+y 1)dy dy 7. 16(x 1)4- + 96(x + 1)d3 + 104(x + 1)2 + 8( + 1) + y =x + 4x + 3. 8. (x2D2 + xD -1)y = xm. 9. (x2D2 - 3 xD + 4)y = x-. 10. (x4D4 + 6 x3D3 + 9 x22 + 3xD + 1)y = (1 + logx)2. dx3 dx2 dx 11. x4 dY + 2 x3 2 jY+ xy-=1. 12. (2D + 3 xD + 1) ( 13. x2D2 - (2 2 - 1)xD + (m'2 + n2)y = n2x log x. 14. 2d -3dy log x sin (log x) + 1 x14. 2 3x+ - dX2 dlx x 92 DIFFERENTIAL EQUATIONS. [CI. VIII. CHAPTER VIII. EXACT DIFFERENTIAL EQUATIONS, AND EQUATIONS OF PARTICULAR FORMS. INTEGRATION IN SERIES. 72. In this chapter linear differential equations that are exact will be first discussed, and then equations of certain particular foriris will be considered. Some of the latter corne under some one of the types already treated; but in obtaining their solutions, special modifications of the general inethods can be employed. It often happens that an equation at the same time belongs to several forms; instances of this will be found among the examples in Arts. 76, 77, 78, 79, 80. 73. Exact differential equations defined. A differential equation, f^,y cdy ) f dxan' "" '' -) ~-, is said to be exact when it can be derived by differentiation merely, and without any further process, from an equation of the next lower order ln-1y d7 f(dI - ', -y z' Xdx + c. Exact differential equations of the first order have been treated in Art. 11. 74. Criterion of an exact differential equation. The condition will now be found which the coefficients of a differential equation, d+ d-l+ equation, P+ +- + P,,y+ = P0, () à -x- à ~~~~~~~~~ (1 ~ 72-74.] EXACT DIFFERlENTIAL EQUATIONS. 93 must satisfy in order that it be exact. hle coefficients P,, P,..*, P,, are functions of x; in what follows, their successive derivatives will be indicated by Pl, P'", P('). The first term of (1) is evidently derivable by direct differentiation from PO dL- which is therefore the first term of the integral of (1); on differentiating this and subtracting the result from the first member of (1), there remains (P n- c a~-lY a"-î (P1-PO dclxe-_ + p dXP, +... + Pny. (2) The first term of (2) is evidently derivable by differentiation ron-2y from (P1 - PO') which is therefore the second term of the integral of (1). On subtracting the derivative of this term from (2) there remains dn-2 P d"-aY (P2 — 3 +' + P0.) - +... + P71y. - + n-2 de + -3 The first terni of this expression is derivable from (P2 - P' + P") d-3 which will therefore be the third term of the integral of (1). By continuing this process, the expression Pn — Pn-2'+ +(- 1)n-lPo(n-l) cdy + P, (3) will be reached, the first terni of which is evidently derivable from Pn- P2' + **-+ (- l)-l1P0(-1) Y. (4) Both terms of (3) will be derived from the expression (4), if the derivative of the coefficient of (y) in (4) be equal to P,,, that is, if P- Pn ' +... +(- 1)'Po( -= 0. (5) But if both terms of (3) are derivable from the expression (4), an integral of (1) has now been obtained in the form 94 DIFFERENTIAL EQUATIONS. [Cu. VIII. d'n-\y d..-2.. 3?/ dn../ Po dn- (P1 - Po,) Y<_ Y (P2-1>1t + Po') _Y + "' + P,-1 - P,-2' + ** (- 1)'Po(I1-) }y = CI. (6) Therefore (1) has an integral (6), that is, (1) is exact if its coefficients satisfy condition (5). 75. The integration of an exact equation; first integrals. If the second member of (1) Art. 74 be f(x), and condition (5) be satisfied, the second member of (6) will beff(x) dx + c. If equation (6) Art. 74 is also exact, its integral can be found in the same way; and the process can be repeated until an equation of the second or higher order that is not exact is reached; in some cases, this process can be carried on until a value of dy, or of y is obtained. Equation (6) is called first dx integral of (1) Art. 74. It is easily proven by means of Arts. 3, 4, and Note C, p. 194, that an equation of the nth order has exactly n independent first integrals.* Sometimes equations that are not linear can be solved by the 'trial method' employed in the case of (1) Art. 74, for instance, Exs. 6, 7, below. Ex. 1. Solve x Y +(x2-3) d2+4 4 +2y =0. This is neither a homogeneous linear equation, nor one having constant coefficients. In it, PO = x, P1i x2 - 3, P2 = 4 x, P3 = 2; and the condition that it be exact is satisfied by these values. Integration gives Y + (x2 - 4)Y + 2 xy - c. The condition under which this equation is exact is also satisfied; integrating again, dy x- + (x2 - 5) y = clx + c2. See Forsyth, Di til Eqtio Arts. 7 8. * Sec Forsyth, Diffeîrential Eqpations, Arts. 7, 8. ~ 75.] EXAMPLES 95 This is not exact, but it is a linear equation of the first order, and hence solvable by the method of Art. 20. The solution is _ _x2 X2 e2y C e2 ye2 X5 = C l 5 dx - c x-dx + C3. d2y dy Ex. 2. Solve x5 + +3x3 -(3 6 x)xy = 4 + 2x 5. dx2 dlx Thle coefficients of this equation do not satisfy condition (5) Art. 74; and hence it is not exact. However, an integrating factor xm can be deduced, which will render it an exact differential equation. Multiplication by x'" gives dx5+m -T + 3 x3+'- + (3 - 6 x)x2+my - (x4 + 2 - 5)m. dx2 d'dx Application of condition (5) Art. 74 to the first member, shows that for that condition to hold, m must satisfy the equation (m + 2) (m + 7)xm+3 - 3 (mn + 2)xm+2 = 0; that is, m must be euual to - 2. On using the factor -, the original equation becomes + 3 x +(3 -- 6x)y = X2 + -- d x which is exact. Integration gives the first integral X3y + 3x(1 - x)y = - +2 logx + dx x linear equation of the first order. Ex. 3. Solve x 2 +2y = 0. dX2, dx d'2y dy Ex. 4. Solve + 2 exy + 2 ey = 2. dx2 dx - day dy Ex. 5. Solve x d + 2 + 3 y = x. Ex. 6. Find a first integral of d - - x2yd = xy2. Ex. 7. Solve xd2 +(x - y) 3 2 = 0. 96 DIFFERENTIAL EQUATIONS. [CH. VIII. dny 76. Equations of the form d —f(x). This is an exact differential equation. Integration gives - r dx —L= j 1(x) dx + c; a second integration gives dX-2y _ f (x) (clx + cx + c,; by proceding in this way, the complete integral y-= f.. f (x)(dx + an-l + a,-2 +... + a~-x + a is obtained. Ex. 1. Solve d-=xe. dx3 d2Y e dxy Integrating, d=xe - e" + Ci, dy xex - 2 ex + cax + C2, dx y = xe - 3 ex + X2 + c2x + C3. This equation could also have been solved by the method of Chap. VI. Ex. 2. Solve d= XM dxx 2d4y/ Ex. 3. Solve x2 4 + 10. d2x4 Ex. 4. Solve =2 sin. dx2 - cldy 77. Equations of the form d Yf(y) An equation of the form d'y dx f(Y)' which in general is not linear and is not an exact differential equation, can be solved in the following way: dy Multiplication of both sides by 2 d- gives ~2 dx2 dx' dy d2y dy. 2d i-P -- 2f(y) d; ~ 76-78.] EQUATIONS NOT CONTAINING y. 97 integrating, (l) =2 Jf (y) dy + c1. dy From this, = dx, 2jf(y) dy + c whence j - d — r+ C2. 2ff(?. )y + c, I Ex. 1. Solve d2y + ay = O. dx2 dy dy d2y -ady. Multiplying by 2 dx, 2 dx dx2= 2 integrating, (c - -a22 + C = (2 - y2 on putting a2c2 for c1; separating the variables, - = adx,.\c2 - y2 and integrating, sin-~1 = ax + c2, hence, y = c sin (ax + c2). The given differential equation is linear, and y can be obtained directly by the method of Art. 52. The roots of the auxiliary equation being I ia, the solution is y = ci sin ax -+ c2 cos ax; that is, y cl sin (ax + c3). Tlhis equation is an important one in physical applications. Ex. 2. Solve d=Y dxy a2 Ex. 3. Solve -2 = 0. cdx2?/2 dS'y Ex. 4. Solve - d ay = 0. dx'- 78. Equations that do not contain y directly. The typical form of these equations is f\(d'?J '" ' ' = )o (1)y 98 DIFFERENTIAL EQUATIONS. [CH. VIII. Equations of this kind of the first order were considered in Art. 26; and those of Art. 76 also belong to this class. dy d2y dp dny d't-p If p be sbstituted for, then d d and (1) takes the form f,, -. l-o=0 an equation of the (n - 1)th order between x and p; and this may possibly be solved for p. Suppose that the solution is dy (); dx then y = F (x) dx + c. If the derivative of lowest order appearing in the equation dry dry be put r- =p, find p, and therefrom find y by Art. 76. dxr' dxr Ex. 1. Solve x2 dy 4 x d+6 4. Putting p for Y, this becomes d2 x2 dp 4x xP + 6p = 4; dX2 dx integrating, p = C1X2 + c2x3 + 3, whence y = ax3 + bx4+ f x + c. Ex. 2. Solve d = 1 + ( Ex. 3. Solve (1 + x2) + 1 + (y) = Ex7. 4 Solve 2 xq - - do yX ) - a 79. Equations that do not contain x directly. The typical form of these equations is f(dny.dy 0) o 0 (1) *'dx' nl dx^) l ~ 79, 80.] EQUATIONS NOT CONTAINING x. 99 Equations of this kind of the first order were considered in Art. 26; and those of Art. 77 also belong to this class. dy If p be substituted for d, then dxy d2y dp dy,d21 d dp'~\2 -d = p, p2 + p ), etc.; dx dy dxl dy' \cdY) and (1) will take the form f( dyn *, py) =O, an equation of the (n - 1)th order between y and p which may possibly be solved for p. Suppose that the solution is p =f(); then the solution of (1) is r dy J f(y)= X + C. Ex. 1. Solve d2 + a (Y) 0. ydx2 (dx) Ex. 2. Solve yd2y-+(dY = 1. dx2 \dx/ Ex. 3. Solve y — - YlogY Ex. 4. Solve d2y + 2d + 4Y = 0 by the method of this article, and by that of Art. 78. 80. Equations in which y appears in only two derivatives whose orders differ by two. The typical form of these equations is (dny dn-2Y. x f:~,d~x,,dn _,, 100 DIFFERENTIAL EQUATIONS. [Cn. VIII. d n-2ny d"?'y cdq If q be substituted for dY then d^= d -; and (1) becomes cd ni dX2 dXcl f 2q x) =o f w hh, is=0, from which q, that is, d,_>, may be found. Suppose that the solution is cl' 2y q d-x- (2 x) then y can be found by the method of successive integration shown in Art. 76. Ex. 1. Solve d Y + a(-2 = 0 both by this method and that of Chap. VI. dX4 (ixJ Ex. 2. Solve 1- 2 Y = ea both by this niethod and that of x. 2 Sl 3 Chap. VI. Ex. 3. Solve x2 d4y + d dX4 dX2 81. Equations in which y appears in only two derivatives whose orders differ by unity. The typical form of these equations is e(C bL y d \.... f \^? t^ x~ )-= ~ (o dn-ly 7yd'y_ dq If q be substituted for -d, then d x= dl; and (1) becomes f d qx dx \^clx ' /)=, an equation of tie first order between q and x; its solution will give the value of q in terms of x. Suppose that the solution is dnF-ly ) =d_-F); 81, 82.] INTEGRATION IN SERIES. 101 then, from this relation, by successive integration, the value of y can be deduced. Ex. 1. Solve a2y X. dx2 dx On putting q for -y, this becomes dx d a2 Q = X; dy integrating, aq - ad = x-/x2 + c2; dx integrating again, ay = - [xCx + c2 + c2 log (x + vx2 + c2) + C2]. Ex. 2. Solve adi2 (d d2y dy Ex. 3. Solve x +Y -0. dx2 ldx Ex. 4. Solve d - 2. dx3 dx2 82. Integration of linear equations in series. When an equation belongs to a form which cannot be solved by any- of the methods hitherto discussed, recourse may be had to finding a convergent series arranged according to powers of the independent variable, which will approximately express the value of the dependent variable. For the purposes of this article it is assumed that such a series can be obtained. Suppose that the linear equation PO z + P + +,_l+ * + Py= (1) can have a solution of the form y = A0x' + A1xTm + A2xm2 +... + * +..., (2) where the second member is a finte suii or a convergent series for some value or values of x. Concerning this series three things must be known: nainely, the initial term, the relation * See Note B. Also Forsyth, Differential Equations, Arts. 83, 84. 102 DIFFERENTIAL EQUATIONS. [CH. VIII. between the exponents of x, and the relation between the coefficients. The law for the exponents will be apparent on substituting xm for y in the first Inemlber of the given equation. Suppose that the expression obtained by this substitution is fi (m) X 2' + f2 (1) X'. (3) In general (3) will contain more than two ternis; in the case of the equations in Art. 66 it contains onrly-one term. Under the supposition just made, the successive differences of the exponlents of x in the series sought must evidently be nir -m1'; this common difference will be denoted by s. Solution (2) may now be written y = Aoàx + A1xms+ +.+. + Ar_Xm~+(r- 1 + Ar t+rs +.*, (4) or simply y= 2 ArX+rs r=O Substitution of this series for y in (1) will give, in virtue of (3), Aof, (ni) xm' + of2 (n) Xm'+s + Alf, (m + s)x ++ Af2(m + s) 'm+2s + * m* + Ar-lf, [m + (r - 1) s] xn+(r-s = + Arf12 [ni + (r - 1)s] X +rs + Arf (n + rs) X+~rs + Af2 (nm + rs) x'+(r+1)s Since equation (5) must be an identity, the coefficients of each power of x therein must be equal to zero; hence.fi(m)=0 (6) and Arf (m + rs) + 4df2 [m + (r - 1) s] = 0. (7) The roots of (6) give the initial exponents of series that will satisfy (1); and equation (7) shows that ~ 82.] INTEGRATION IN SERIES. 103 Ar- fi + (r -+r s) r which is the relation between successive coefficients. The difference between the exponents in (3) might have been taken, mi - m'I or — s; in this case, the resulting series would have had their powers in reverse order to those of (4); and the initial terms would have been found by solving f2 (m)= 0. In determining the initial power of x for an equation of the nth order, that coefficient in (3) which is of the nth degree in m must be put equal to zero, since there must be n independent series in the general solution. If both fi (m) and f2(i) are of the nth degree, two sets of series can be derived, one in ascending powers and the other in descending powers of x. If the expression (3) have another term f3(nx) X'"n, the terms of the series can be successively deduced, but the process will be much more tedious. This method can also be employed in the case of non-linear equations, but more than a very few terms can be calculated only with difficulty. The equations previously considered can of course be integrated in series; Ex. 2 below will illustrate this. The procedure when the second member of (1) is not zero will be made clear in the first example below. Ex. 1. Integrate (1) xd + xd + y = x-1. dx2 dx First find the complementary function. The substitution of x-n for y in the first member gives (2) m(m - 1)xm+2 + (m + 1)xm; whence s - 2, and m = O or 1. r=oo The substitution of " A.rX-2 for y gives r=O [(m - 2 r) (m - 2r - 1)ArXm-2r+2 + (m - 2 1 1)ArXm-2r]= O. The coefficient of Xm-2r+2 must vanish; therefore (n - 2 r) (n - 2 r - 1) Ar + (1- - 2 + 3)Ar-i = 0, 104 DIFFERENTIAL EQUATIONS. [CH. VIII. h m - 2 r + 3. hence (3) Ar= -.(m - 2 r) (Ç - 2 r - 1) which is the relation between the coefficients. For m = 0, A 2-3 A. 2r(2 r+ 1) hence A= - Ao 2.3 A2= Ai - Ao, 4 5 5! A3 = - 2 =- Ao, etc.; 6.7 7! the corresponding series is 1 X-2_ 1 X-4 _. 1 -63 - 1 3 5xs 3! 5! 7! 9! For m = 1, Ar- 2r- 4 A-l 2r(2r - 1) hence A1 =- Ao = - Ao, 2(2- 1) 4-4 A2 = Ao = 0, 4(3- 1) and A3, A4, *.. are each equal to zero; the series in this case is finite, being x —. Hence the complementary function is A(1_ tx_2- 1 -4- 1 ) -... i, 3! 5!! x In order to find the particular integral, substitute Aozx for y; then must nm(m - 1)AoX+2 = X-1; comparison of the exponents shows that m =- 3; and hence Ao = 12 For m - 3; the relation (3) between the coefficients becomes Ar = Ar (2 r + 3)(2r + 4) hence A1= 2 Ao, A2 2 Ao, A3= -Ao.., 5.6 5.7.6.8 5796810 ~ 83.] EQUATIONS OF LEGENDRE, BESSEL, ETC. 105 and the particular integral is x-a 1 + 2 24 12 5( 6 ~ 5~7~78 ~ 6! 8! that is, 2x-3(1 +- 2 2. 4x-4..) Ex. 2. Show by the methocl of integration in series, that the general d2y solution of -- + y = 0 is A cos x + B sin x. Ex. 3. (2x2 +1) d 2y = 0. Ex. 4. 2 2 d x + (1 X2)y = x2 x2 dx x2)y x2. d2Y - dy Ex. 5. (1 - x2) x2 + n(n + l1)y=0. 83. Equations of Legendre, Bessel, Riccati, and the hypergeometric series.* A fuller discussion of integration in series than is here attempted is beyond the limits of an introductory course in differential equations. The purpose of Art. 82 has merely been to give the student a little idea of a method which is of wide application; and which is used in solving four very important equations that often occiur in investigations in applied mathematics, - the equations of Riccati, Bessel, Legendre, and the hypergeometric series. Johnson's Differential Equations, Arts. 171-180, discusses the methods to be followed when two roots of (6) Art. 82, become equal, the corresponding series then being identical; and when two of the roots differ by a multiple of s, one series then being included in the other; and when a coefficient Ar is infinite. The equations referred to above, and references to be consulted concerning them, are as follows: t Legendre's equation is (1 - X2) d2 2 x dy n(n + l)y = 0, * In connection with this article, the student is advised to read W. E. Byerly, Fourier's Series and Spherical Harmtonics, Arts. 14-18. t Adrien Marie Legendre (1752-1833) was the author of Elements' of Geometry, published in 1794, the modern rival of Euclid. He is noted for his researches in Elliptic Functions and Theory of Numbers. He was the creator, with Laplace, of Spherical Harmonics. 106 DIFFERENTIAL EQUATIONS. [CH. VIII. or d xx ( x) } + (n+ l) =0, where n is a constant, generally a positive integer. See Ex. 5, Art. 82. (Forsyth, Diff. Eq., Arts. 89-99; Johnson, Diff. Eq., Arts. 222-226; Byerly, Fourlier's Series and Spherical Iiarmonics, Arts. 9, 10, 13 (c), 16, 18 (c), and Chap. V., pp. 144-194; Byerly, Harmonic Functions (Merriman and Woodward, Higher Mathematics, Chap. V.), Arts. 4, 12-17.) * Bessel's equation is 2 d2y +.0dy + x + (X2 - n2)y = 0, in which n is usually an integer. (Forsyth, Diff. Eq., Arts. 100-107; Johnson, Diff. Eq., Arts. 215-221; Byerly, Fourier's Series, etc., Arts. 11, 17, 18 (d), and Chap. VII., pp. 219-233; Byerly, Ifarmonic FZnctions, Arts. 5, 19-23 of Chap. V. in Higher Mathematics; Gray and Matlews, Bessel Functions and their Applications to Physics; Todhunter, Laplace's, Lamé's, and Bessel's Functions.) t Riccati's equation is dy + by2 = cxm, dxy to which forin is reducible the equation x d - ay + by2-= cx. The clx latter equation is integrable in finite terms when n 2 a, or when n + a 2 n is a positive integer. Riccati's equation can be reduced to a linear form, d2u but of the second order, d a2xm- 0. dx:2 (Forsyth, Diff. Eq., Arts. 108-111; Johnson, Dif. Eq., Arts. 204-214; Glaisher, lMemoir in Phil. Trans., 1881, pp. 759-828.) * Frederick Wilhelm Bessel (1784-1846) may be regarded as the founder of modern practical astronomy. In 1824, in connection with a problem in orbital motion, he introduced the functions called by his nmne which appear in the integrals of this equation. t Jacopo Francesco, Count Riccati (1676-1754) is best known in connection with this equation, which was published in 1724. He integrated it for some special cases. ~ 83.] EXAMPLES. 107 * The differential equation of the hypergenometric series is d2y y - (a + P 1)x dy al dx2 x(1 - x) dx x( 1-x)y This equation has the hypergeometric series 1+ a ( x a(a + (P +1) x2 +... 1.-y 1.2.y. +1 usually denoted by F(a, 3, y, x), for one of its particular integrals; and has a set of 24 particular integrals, each of which contains a hypergeometric series. (Forsyth, Diff. Eq., Arts. 113-134; Johnson, Diff. Eq., Arts. 181-203.) EXAMPLES ON CHAPTER VIII. 1. Show that the following equation is exact and find a first integral. (y2 + d/ + 2(y + x) (dy 2 x + y 2 d2Y a2 dy x2 dx2 x(a2 - x2) dx a(a2 - x2) d3 y d2y y 3. (1 +x +x2) +(3 + 6) + 6 -=0. dx,3 dx2 dx4. Find a first integral of x3 d 4 2 d + x(x2 + 2) d + 3x2y = 2x. Fndx3 + dx2 dix d4y a d2Y O 5. a2y -. dx4 dX2 d (y\2 d2y (dy\2 +, (d2?\2y 6. < ix) - Y dX2 I d + a 2 * * This is also called the Gaussian equation, and the series, the Gaussian series, after Karl Friedrich Gauss (1777-1855), who is regarded as one of the greatest mathematicians of the nineteenth century. He is especially noted for his invention of a new method for calculating orbits, and for his researches in the Theory of Numbers. It was Euler (see footnote, p. 64) who discovered the series and set forth its differential equation; but Gauss made important investigations concerning the series, and showed that the ordinary algebraic, trigonornetrical, and exponential series can be represented by it. (For illustrations of the last remark, see Johnson, Differential Equations, Ex. 1, p. 220.) 108 DIFFERENTIAL EQUATIONS. [Ci. VIII. 7. (x-x)+(8 3)y14x 4. 7 (X3-X) dd Y+(8 x2- 3) + 14x +4Y:Xdx8 dxd + )x d X/2 d2y dy ( 1dy 31lg d2Y -dY 8. dy +ly+ O. 10. sinx cosx 2 ysin x - 14. d+si+\. d= dx3 d dx2 d /39. (1- x2)C-f dy 20. Find three independent first integrals of -_ z/f(x). 212. 11. x - X 0.d2y a d2y d12 -=X' 13. y(l - logy) ) V ]x) =Y0. d3dy dy e:. 14. 3 sin2 x. 15. + [:+ ex. d3y d2y Xdy 16. ~+ cosx — 2sin x y cos jÏX3 + X2 dx o sin2x. d2y d'y dy d2Y 17. sin2x ax 2 Y. 18. a 19. -' a. cEcr;2 21 9. ' - - — a. 20. Find three independent first initegrals of:- f(x). 21. j2y a2 x'2= dx ~ 84, 85.] EQUATIONS OF THE SECOND ORDEIR. 109 CHAPTER IX. EQUATIONS OF THE SECOND ORDER. 84. There are other methods of solution, different from those shown in the last three chapters, which are applicable to some equations of the second orcler; Arts. 85-89 will be taken up with an exposition of three of these methods. If a differential equation is not in a form to which any of the methods already described apply, it may be possible to put it in such a form. The very important transformations of an equation that can be effected by changing the dependent or the independent variable will be discussed in Arts. 90-92. Art. 93 will contain a synopsis of all the methods considered up to that point which may be employed in solving equations of the second order. 85. The complete solution in terms of a known integral. A theorem of great importance relating to the linear differential equation of the second order, is the following: If an integral includes in the complementary function of such an equation be known, the complete solution can be expressed in terms of the known integral. Suppose that y = y, is a known integral in the complementary function of dX2 P + Qy = X; ( then the complete solution of (1) can be determined in terms of y,. Let y =- y1 110 DIFFERENTIAL EQUATIONS. [Ci. IX. be another solution of (1); v will now be determined. On substituting y1v for y in (1), it will become dv +(p 2 dy1 dv=X. dà2\ y, ddx ) dx (2) since, by hypothesis, d2y- + P dy, + QY1 =. dx-2 dx dv On putting p for -d, (2) becomes dx' d(p (~2 dyi x;( d+(x +Yi dx)P=; (3) and this equation, being linear and of the first order, can be solved for p. On using the method of Art. 20, the solution is found to be dv cle-SPdx eS PdX r P = — = -- + le| y 1e'X P; dx yl2 Y1 J whence, integrating, v = C2 + cl e- x dx + ( e jy lePX(dX)2. J Yi2 J Yi, Therefore another solution of (f) is.e-PdPa~ pe-jPd'a ~r y = yi = ci2Y1 Y e d + cy e 7f. yIefPdxX ( yx)'2.( (4) This includes the given solution y = y,; and, since it contains two arbitrary constants, it is the complete solution. From the forni of the solution (4), it is evident that the J- f-Pdx second part of the complementary function is yJ 2 - dx, an that the particular integral is y Si and that the particular integral is y - 1 Y le SPdX(dx)2. yi J ~ 86, 87.] SOLUTION FOUND BY INSPECTION. 111 86. Relation between the integrals. It is easily shown, that if y = y,, y = Y2 be two independent integrals of d2y dy d'y ~ P dj + QY = 0, dx' 2 dx then Yi1 ctdy2 dy, = ce-f PdX dx (l (See Forsyth's Dig. Eq., Art. 65; Johnson's Dif'. Eq., Art. 147.) It may also be remarked in passing, that the deduction of (3) Art. 85 froin (1), when an integral of the latter is known, is an example of the theorem: that, if one or several independent integrals of a linear equation be known, the order of the equation can be lowered by a number equal to the number of the known integrals. (See Forsyth's Diff. Eq., Arts. 41, 76, 77.) 87. To find the solution by inspection. Since the complete integral of (1) Art. 85 can be found if one integral in its complementary function be known, it is generally worth while to try whether an integral in the latter can be determined by inspection. Ex. 1. Solvex + (1 -x) -y = ex. dIX2 (lx Here, the sum of the coefficients being zero, ex is obviously a solution of xd +(1 - x) -y = O. Substitution of vex for y in the original equation gives d2v dv X + (1 + x) -1; x (l+2 (~l this, on substituting p for d, becomes dx, + (1 + x)p =, a linear equation of the first order. Its solution is dxv e- 1 dx x- x 112 DIFFERENTIAL EQUATIONS. [CH. IX. hence v = log x -+ cIxle-xdx + C2; and therefore the complete solution is y ex log x + clexx le-xdx + c2ex. Equation (4) Art. 85 miight have been used as a substitution formula, but it is better to work out each example by the same general method by which (4) was itself derived. Ex. 2. Solve 2 - x + xy x. clx2 dx [Here, y = x is obviously a solution when the second member is zero. A solution can often be found by an inspection of the terms of lower order in the equation.] Ex. 3. Solve (3-x) d- (9 -4x) d+ (6- 3 X) =O. àl"! i 1dx Ex. 4. Solve x d2 y -, get Ex. 4. Solve x -+ x-+ - y = 0, given that x + - is one integral. 88. The solution found by means of operational factors. Suppose that the linear equation of the second order d(2y dy PO -_ + Pl + -P2y — '5, Clx2 'clx is expressed in the form f(D)y = x. Sometimes f(D) can be resolved into a product of two factors Fi(D) and F,(D), such that, when Fi(D) operates upon y, and then F,(D) operates upon the result of this operation, the same result is obtained as when F(D) operates upon y. This may be expressed symbolically, f (D)y = F2(D) F,(D)y} i or simply,- f(D)y = F,(D)F1(D)y, it being understood that the operations indicated in the second member of the last equation are made in order from right to left. Factors of this kind have already been employed in dealing with linear equations with constant coefficients, and with ~ 88.] SOLUTION FOUND BY FACTORING. 113 the homogeneous linear equations, Arts. 53, 55, 67, etc. With the exception of the classes of equations just mentioned, the factors are generally not commutative; this can be verified in the case of the examples below. If one of the integrals be known, its corresponding factor is known, and the second factor can be determined by means of the equation and the known factor. For instance, if y = e be an integral of the given equation, then (D - 1)y is the corresponding factor; if y = x be an integral, (xD - 1)y is the corresponding factor. The following example will make the method of procedure clear. d2y dy Ex. 1. Solvex d + (1 - x) -y = e. dx2 dx This equation, wliich is Ex. 1, Art. 87, when written in the symbolic form, is [xD2 + (1 - x)D - 1]y = e; (1) on using symbolic factors, it becomes (xD + 1)(D- )y-= e. (2) [These factors are not commutative, for (D - l)(xD + 1)y on expansion gives {xD2 + (2 - x)D - ly]. Let (D - l)y = v, (3) and (2) becomes (xD + l)v = ex; whence, v = cx-1 + exx-1. Substitution of this value of v in (3) gives (D - l)y = cx-1 + exx-1; whence, on integrating, y - ce + ce e —x xclx + e logx, the solution found in the last article. Ex. 2. Solve Ex. 3, Art. 87, by this method. Ex. 3. Solve 3 x2 d+(2- x2) -4 =0. xx + (2 clx- - Ex. 4. Solve 3 x2 + (2 + 6 x - 6 x) -4 = 0. C-X2 dx 114 DIFFERENTIAL EQUATIONS. [CH. IX, 89. Solution found by means of two first integrals. It follows, from a statement made in Art. 75, that a linear equation of the second order has two first integrals of the first order. If these integrals be known, then ~- can be eliminated becdx tween them; the relation thus found between x and y will be a solution of the original equation. Another method of solution that can be used in the case of the linear equation of the second order is the "method of variation of parameters." * As most of the equations solvable by it are solvable in other ways, and as it is rather long, it will not be given here. (See Johnson's Diff. Eq., Arts. 90, 91; Forsyth's Diff. Eq., Arts. 65-67.) Ex. Solve a2 ( ) =1 d2Y ) by means of the first integrals. On putting dy=p, d2=p and integrating, there appears a first On putting =Pl dx2 =x1 x, integral p +x /l+p2 = e". d2y dp On substituting for - its equivalent expression p, and integrating, dX2 dy another first integral is obtained, a2p2 = (y + c2)2 - 2. The elimination of p between these first integrals gives the solution y + c2 = a cosh + ca 90. Transformation of the equation by changing the dependent variable. Sometimes an equation can be transformed into an integrable type by changing the dependent variable. If any linear equation of the second order, dx2y d * This method is due to Lagrange. ~ 89-91.] CHANGE OF THE DEPENDENT VARIABLE. 115 be taken, and ylv be substituted for y therein, yi being some function of x, (1) will be transformed into dv 2 (P d d 1 d gy pv + O ) x dx P- y, dx y~ dx2 -dx Yi which has v for its dependent variable. This equation may be written _v p dv X d+pdV+ +Q1v=-, (3) dx' cd Yi where P = P+2 d (4) Y1 dx and Q = + P + Q (5) Any value desired can be assigned to P, or Q, by means of a proper choice of y,. Thus, Q, will be zero if y, be chosen so that 1 + QY a_ - P - Qyl = 0; this is what was done in Art. 85. Again, P,, the coefficient of the first derivative in (3), can have any arbitrary value assigned to it; but then y, must be chosen so as to satisfy (4); that is, y1 = eS(P-P)dx. (6) 91. Removal of the first derivative. In particular, it follows from (4) or (6) Art. 90 that P, is zero if Yi = e-fPd. On substituting this value of y, in the coefficient of v in (2) Art. 90, this coefficient becomes Q - dP p 116 DIFFERENTIAL EQUATIONS. [Cil. IX. Therefore the differential equation (1) Art. 90 of the second order is transformed into a differential equation not containing the first derivative, by substituting ve-P~ldx for y; and the transformed equation is -9 + Qlv-X1, (1) dx2 where Q1== Q -d 1- -2, and = XelPd. The new equation (1) may happen to be easily integrable. Transforming (1) Art. 90 into the form (1) is called "r emoving the first derivative." The student should mnemorise the above values of the new Q1 and X1, in terms of P, Q, X, for then he can immediately write cown the new equation in v, without the labour of making the substitution in the original equation and reducing. It may be remarked in passing that this removal of the term next to the second derivative is merely an example of the general theorem, that the coefficient of the terni of (n - 1)th order in a linear equation dny -PS-ly x + Pl - +... + P2y = ix, dx" dx?' can be removed by substituting yv for y, where -iSPicix Yi = e l" The reader can easily verify this by making the substitution. (See Forsyth's Diff. Eq., Art. 42.) Ex. 1. Solve d y - 1 ) 1 6 ) Here P=x3 Q= - 4 6 and henceyl = e-JPdx e-A 4x3 6 3 ~ 92.] CHANGE OF THE INDEPENDENT VARIABLE. 117 If the second term be removed, Q1=QdP lp2= 6 dx - and hence the transformed equation is d2v 6 v dx2 X2 the solution of which is V = C1X3 + C Hence the general solution of the given equation is y= y =e —lî(cx3+ ~2 Ex. 2. Solve 4x2d + 4 X5 2+ (x8 + 64+4)y =0. 'd y dy Ex. 3. Solve dy + 2 tan x + 5 y 0. dx2 +x Ex. 4. Solve 2 2(2 + ) + (2 + 2x + 2)y =0. 92. Transformation of the equation by changing the independent variable. An equation can sometimes be transformed into an integrable form by changing the independent variable. Suppose that d2 + P y = X (-) is any linear equation of the second order, and that the independent variable is to be changed from x to z, there being some given relation, z f(x), connecting x and z. dy __dy dz and d Y d= y (dz,X + d 2 Since a — = am ( d x' cdx dz dx' dx' dz-,dx) dz dx2' d"y dy (1) becomes d2 + Pd + Q= X, (2) d + d d'z + P dz where P = dl dx, and X, = (- (3) dxz)2 (zd)2 118 DIFFERENTIAL EQUATIONS. [CH. IX. P1, Qi, Xi, as just expressed, are functions of x; but can be immediately expressed as functions of z by means of the relation connecting z and x. Any arbitrary value can be given to PI; but then z must be so chosen that it satisfies the first of equations (3). In particular, P1 will be zero if d2- + P -c 0, that is, if z = e-SPdxdx. dx2 dx Again, the new coefficient Q1 will be a constant, a2, by virtue of the second of equations (3), if a2 dZ = Q, that is, if az = VQ dx. Ex. 1. Solve d + 2 dy _. dx2 x dx X4 Find z, such that (dz 2= a2 solving, a dx/ x; Change of the independent variable from x to z will now give d'2Y+ y dz2 and this has for its solution y = A cos z + Bsinz. Hence the solution of the given equation is a a y = ci cos- + c2 sinx x d2y. dy Ex. 2. Solve d + cot x d- + 4 y cosec2 x = 0. Ex. 3. Solve x d- + 4 X3 = 5. dxz'2 (lx Ex. 4. Solve x6 d+ 3x5 a2y 93. Synopsis of methods of solving equations of the second order. This article is merely a synopsis of all the methods discussed thus far in the book that are employed in the solution of equa ~ 93.] SYNOPSIS OF METIODS OF SOLUTION. 119 tions of the second order. Several of these methods may be suitable for solving the same equation. The references are to the chapters and articles where the methods are described. The student is advised to select a few equations of the second order from the articles referred to, and to solve each one in two or more different ways. An equation of the second order may be (a) linear with constant coefficients, [Chap. VI.]; (b) a homogeneous linear equation, [Chap. VII.]; (c) an exact differential equation, [Arts. 73-75, 76]; (d) an equation that does not directly contain the dependent variable, [Arts. 76, 78]; (e) an equation that does not directly contain the independent variable, [Arts. 77, 79]; (f) in the form d2y =/f [Art. 77]; (g) an equation, one of whose integrals is known or is easily found by inspection, [Arts. 85, 87]; (h) factorable into symbolic operators, [Art. 88]; (i) an equation of which two first integrals can be easily found, [Art. 89]; (j) an equation that can be integrated in series. [Art. 82]. If the equation is not in an integrable form, it may be put in such a form by (a) so changing the dependent variable, that (1) the coefficient of the first derivative will have an assigned value [Art. 90]; or that (2) (in particular), this coefficient will be zero [Art. 91]; (b) so changing the independent variable, that (1) the equation will be transformed into the linear form with constant coefficients, or into the homogeneous linear form [Art. 71]; 120 DIFFERENTIAL EQUATIONS. [Ci. IX. or that (2) the coefficient of the first derivative will liave an assigned value, and, in particular, the value zero [Art. 92]; or that (3) the coefficient of the variable will have an assignec value, and, in particular, be a constant [Art. 92]. EXAMPLES ON CHAPTER IX.. d+ 2l y 3' -- + (2 +.dx'-: x cd -1- n~'' X six 2. dy + + t2J = 0 4. (1 X') d2 + 3 x +y = 0. dx2'~xd~U 4. d2 d 5. (x-3) d2- (4 x - 9) d + 3(x - 2) y = O, ex being a solution. 6. J 2 bxdy + b2X2y 0. cdx'2 CiX 7. +4x- + 4 xy = 0. d2y dx 8. x d -(2 x - 1) + (x - l)y = O, given that y = ex is a solution. 9. (1 -x 2) d+ x -xJ = -X(1- 2) d2yi dy 10. (x sinx+cosx) c -x cos x - +y cos x=O, of vhich y-=x is a solution. 11. X2+2 I-2 d + 2J = 0. dx3 dx2 dx 82!I. dy 12. (1 - 2) d -- x- - a2y O, of which y = Ceasin"- is an integral. cix" (lx d"y.dy d 13. d -Y + x dy =f(x). dx2 dx 14. X2 2 2x(l + ) d +2(1 + x)y = x3. dx'2 dx dzy a2 dy x2 15. (a2-x2)-X + l = 0 X2x dx c- y= d2y dy d218yi (cldy\ 2 16. (x3-x) 2-lf-+ +3 nx3y -. 18. y + -- + - cd x (dx dx' ( x/ 2 d2y dy 2- 0. 1. d2y ' d 17. x22y /dI \y- 19. 2 x4 2 - +n2y = O. dXy2 dxi ' dx2 d ~ 94, 95.] GEOMETRICAL APPLICATIONS. 121 CHAPTER X. GEOMETRICAL AND PHYSICAL APPLICATIONS. 94. Chapter V. was clevoted to geometrical and physical applications; but the choice of problems for that chapter was restricted by the condition that a differential equation of an order higher than the first should not be needed in determining their solution. The practical problems now to be give are of the same general character as those already set; but ir order to obtain their solution, equations of orders higher than the first may be required. 95. Geometrical Problems. The following can be added to the geometrical principles and formulae given in Art. 42. The radius of curvature in rectangular co-ordinates is 3 d2y dx2 If the normal be always drawn towards the x-axis, both it and the radius of curvature at any point on the curve are drawn in the same direction when y and dly at th point are opposite in sign, and they are drawn in opposite directions when y and dy agree in sign. This will be apparent on drawwhen y and 12 agree in sign. This will be apparent on drawd7X2 ing four curves, one concave upward and one concave downward, above the x-axis, and two similar ones below this axis. 122 DIFFERENTIAL EQUATIONS. [CH. X. Ex. Find the equation of the curve for any point of which the second derivative of the ordinate is inversely proportional to the semi-cubical power of the product of the suin and difference of the abscissa and a constant length a; determine the curve so that it will cut the y-axis at right angles, and the x-axis at a distance a from the origin. The first condition is expressed by either of the equations d2y k2, and d2y k2 dx2 (x2 _ a2) dx2 (a2 - 2) Integrating the first equation, dy k2 x dx a2,x2 _ a2 but, by the second condition, O = 0 when x = 0, and hence c = 0; this gives dy k2 X dx ~a2 ax/ -- a2 k2 Integrating, y - - V2 - a2 + c; a2 but, by the third condition, y = O when x = a, and hence c = 0; therefore the equation of the curve reduces to k4x2 _ a4y2 a2k4, the equation of an hyperbola with transverse axis equal to 2 a, and conju2 k'2 gate axis equal to -2 Had the second equation been taken, the equation of an ellipse, k4x2 + a4y2 = a2k4, would have been obtained. 96. Mechanical and physical problems. The following can be added to the mechanical principles and formula given in Art. 48; s, v, x, y, r, 0, t, have the same signification as before. dJ2s = the acceleration of a moving particle, at any point in its dt" dt path. d2x c- = the component of the acceleration parallel to the x-axis. dt2 ~ 96.] MECHANICAL AND PHYSICAL APPLICATIONS. 123 d2_ dt2 =the component of the acceleration parallel to the y-axis. d2s [(d2x)2 d2y1 dt2 L t2 t2 jd d2F d-= the angular acceleration about a fixed point. dt2 The force acting upon a particle is equal to the product of the mass of the particle by the acceleration of the motion of the particle due to the force.* An attracting force causes negative acceleration, and a repelling force causes positive acceleration. Ex. 1. Find the distance passed over by a moving point when its acceleration is directly proportional to its distance from a fixed point, the acceleration being directed towards the point from which distance is measured. Here ds = k2s. dt2 Using the method of Art. 78, 2 ds. 2s 2 ksds dt dt2 dt whence (ds- = k2(a2 - s2), dt where k2a2 conveniently represents the constant of integration. ds Hence kdt /a2 - s2 integrating, sin-' s = kt + b; hence s = a sin (kt + b). ds Also, s can be found directly, without finding d, by the method in Chap. VI. The equation may be written t (D2 + k2)s = 0; * A particular choice of units is presupposed in this statement. 124 DIFFERENTIAL EQUATIONS. [CH. X. hence s - cl sin kt + c2 sin kt; that is, s = a sin (kt + b), as above. Ex. 2. In the case of the simple pendulum of length 1, the equation connecting the acceleration due to gravity and the angle 0 through which tle pendulum swings is I d+go=0. dt2 when 0 is small. Determine the time of an oscillation. Since 1d20 + g e- 0, dt2 i 0 = c1 cos - t + c2 sin t. Let 0 = 00 and c = O when t = 0; applying these conditions, c1 0o, c =- O, and hence ct = 00 cos -t; that is, t =\ cos-1-, g eOo which is the time of swing from 00 to 0. If 0 - o00, t = 7r; hence the time of a complete oscillation from 00 to - 00 and back again is 2 7r - EXAMPLES ON CHAPTER X. 1. Determine the curve in which the curvature is constant and equal to k. 2. Determine the curve whose radius of curvature is equal to the normal and in the opposite direction. 3. Determine the curve whose radius of curvature is equal to the normal and in the same direction. 4. Determine the curve whose radius of curvature is equal to twice the normal and in the opposite direction. 5. Find the curve whose radius of curvature is double the normal and in the same direction. 6. Determine the curve whose radius of curvature varies as the cube of the normal. ~ 96.] EXAMPLES. 125 7. Find the curve whose radius of curvature varies inversely as the abscissa. 8. Find the distance passed over by a moving particle when its acceleration is directly proportional to its distance from a fixed point, the acceleration being directed away from the point from which distance is measured. 9. Find the distance passed over by a particle whose acceleration is constant and equal to a, vo being the initial velocity, and so the initial distance of the particle from the point whence distance is measured. 10. Find the distance passed over by a particle wlen tle acceleration is inversely proportional to the square of the distance from a fixed point. 11. Finc the distance passed over by a body falling from rest, assuming that the resistance of the air is proportional to the square of the velocity. 12. The acceleration of a moving particle being proportional to the cube of the velocity and negative, find the distance passed over in time t, the initial velocity being vo, and the distance being rneasured from the position of the particle at the time t = 0. 13. The relation between the small horizontal deflection 0 of a bar magnet under the action of the earth's magnetic field is dt2 where A is the moment of inertia of the magnet about the axis, 1MI the magnetic moment of the magnet, and H the horizontal component of the intensity of the field due to the earth. Find the time of a complete vibration. 14. In the case of a stretched elastic string, which lias one end fixed and a particle of mass m attached to the other end, the equation of motion is dn2s mg ( sdt2 e where 1 is the natural length of the string, and e its elongation due to a weight mag. Find s and v, determining the constants so that s = So at the time t = 0. 15. A particle moves in a straight line under the action of an attraction varying inversely as the ( —)th power of the distance. Show that the velocity acquired by falling from an infinite distance to a distance a from the centre is equal to the velocity which would be acquired in Inoving from rest at a distance a to a distance a4 126 DIFFERENTIAL EQUATIONS. [CH. X. 16. A particle moves in a straight line from rest at a distance a towards a centre of attraction, the attraction varying inversely as the cube of the distance. Find the whole time of motion. 17. The differential equation for a circuit containing resistance, selfinduction, and capacity, in terms of the current and the time, is d2i R di i 1 f 1 dt2 L dt LC L f(t) being the electromotive force. Find the current i. 18. The differential equation for the above circuit in terms of the charge of electricity in the condenser is d2q- + + = f(t). dt2 Ld t LC L Find the charge q. 19. Solve di + -i = when R2C 4L. dt2 L dt LC di _dt 20. Solve L - + = 0, the differential equation which means that d-t C! the self-induction and capacity in a circuit neutralize each other. Determine the constants in such a way that I is the maximum current, and i = 0 when t = 0. (The given equation, on differentiation, reduces to d- = 0.) dt" L G 21. When the galvanometer is damped, the equation of motion may be written d2 +2 kd+ 2(0 a)=, — 2- $ +w2(o -.)= O, dt2 dt a being the deflection of the needle from the position from which angles are measured, when n its position of equilibrium, the factor k depending on the damping, and w2 on the restoring couple. Find the position of the needle at any instant. (Emtage, Electricity and lcyagnetism, pp. 179, 180.) * 22. Find the equation of the elastic curve for a cantilever beam of uniform cross-section and length 1, with a load P at the free end, the differential equation being EIÎd2=- Px, where I is the moment of inertia of the cross-section with respect to the * Merriman, 3Iechanics of lMaterials, pp. 72, 73. ~ 96.] EXAMPLES. 127 neutral axis, and E is the coefficient of elasticity of the material of the beam. (The origin being taken at the free end of the beam, the x"axis being along its horizontal projection, and the y-axis being the vertical, d- = 0 when x = 1, and y = 0 when x = 0. These conditions are suffidx cient to determine the constants.) - 23. Find the elastic curve when the load is uniformly distributed over the beain described in Ex. 22, say w per linear unit, the differential equation being EId2 wx2 dx2 `2 t 24. Find the elastic curve for the beam considered in Ex. 23, when a horizontal tensile force Q is applied at the free end, the differential equation being EI d2= Qy- wx2. - Merriman, Mechanics of IMaterials, pp. 72, 73. t Merriman and Woodward, Higher Mathematics, Prob. 105, p. 153. 128 DIFFERENTIAL EQUATIONS. [Ci. XI. CHAPTER XI. ORDINARY DIFFERENTIAL EQUATIONS WITH MORE THAN TWO VARIABLES. 97. So far equations containing two variables have been considered. It is now necessary to treat a few forms containing more than two variables. Such equations are either ordinary or particle, the former having only one independent variable, and the latter more than one. In this chapter ordinary differential equations will be discussed. 98. Simultaneous differential equations which are linear. First will be considered the case in which there is a set of relations consisting of as many simultaneous equations as there are dependent variables; moreover, all the equations are to be linear. By following a method somewhat analogous to that employed in solving sets of simultaneous algebraic equations that involve several unknowns, the dependent variables corresponding to the unknowns, there is obtained, by a process of elimination, an equation that involves only one dependent variable with the independent variable; and from this newly formed equation an integral relation between these two variables may be derived. Then a relation between a second dependent variable and the independent variable can be deduced, either (1) by the method of elimination and integration employed in the case of the first variable; or (2) by substituting the value already found for the first variable, in one of the equations involving only the first and second dependent variables and the independent variable. The complete solution consists of as many indepen ~ 97, 98.] SIMULTANEOUS LINEAR EQUATIONS. 129 dent relations between the variables as there are dependent variables. The following example will make the process clear: Ex. 1. Solve the simultaneous equations, (1) + l + 2x + = 0. dt cdt cdt (2) -q-+5q-3 y=0. Differentiation of (2) gives (3) d 5 3dx dy 0. These three equations suffice for the elimination of x and x this dt elimination is effected by multiplying the first equation by - 5, the second by 2, the third by 1, and adding; the result is (4) d +Y = 0. Solving (4), y = A cos t + Bsin t; and substituting this value of y in (2), 3A-+B A-3B x = + B cos t +A -3B sin t. 5 5 By using the symbol D, which was employed in Chap. VI., the elimination can be effected more easily. On substituting D for -d, the given equations become (D + 2)x (D + 1)y = 0, 5x +(D + 3)y= 0. Eliminating x as if D were an algebraic multiplier, (D2 + 1)y = 0, which is equation (4); the remainder of the work is as above. If y had been eliminated instead of x, the resulting equation would have been (D2 + 1)x = 0; whence x = A' cos t + B' sin t; K 130 DIFFERENTIAL EQUATIONS. [CH. XI. substitution of this value in dt which is (1) minus (2), gives 3 B' +A' A B'-3A' y_ 3B' + AI sin t + 3A cos t. 2 2 Substitution is made in (5), because it is easier to derive the value of y from it than from (1) or (2). The second form of solution cones from the first on substituting A' for- 3A + B, and B' for A - 3, the coefficients in the first value of 5 5 x. In general the constants are arbitrary in the value of only one of the dependent variables. Ex. 2. Solve dx —7x+ y=0 dt dy 2x-5y — I dt Ex. 3. Solve d x+2x-3y_ t dt -3 x + 2y e2t J dt Ex. 4. Solve4d +9d +44x+49y=t t dt dt 3 -+7 + 34x + 38 y = et dt dt Ex. 5. Solve -3a -4?y=0' d2 Y+ x+ y=O 99. Simultaneous equations of the first order. Simultaneous equations of the first order and of the first degree in the derivatives can sometimes be solved by the following method, which is generally shorter than that shown in the last article. Equations involving only three variables will be considered; the method, however, is general, and can be applied to equations having any number of variables. ~ 99.] EQUATIONS OF FIRST ORDER AND DEGREE. 131 The general type of a set of simultaneous equations of the first order between three variables is Pldx + Qidy + Rldz = O P2dx + Q2dy + R2dz= 0 J ' where the coefficients are functions of x, y, z. These equations can be expressed in the form dx _ dy _ dz P Q R' where P, Q, R, are functions of x, y, z; for, on taking z as the independent variable and solving equations (1) for - and dz dz dx- Q1R, - Q2R1 dy _ R1P2 - P1R, dz P1Q2-P2Q1 dz P1Q2-P2 whendx d dy dz Q1R2 - Q2R1 R1P2 - R2P P1Q2 - P2Ql' which is the formi (2) above. In what follows, equations (2) will be taken as the type of a set of simultaneous equations of the first order. If one of the variables be absent from two members of (2), the method of procedure is obvious. For example, suppose that z is absent from P and Q; then the solution of dx _ dy P Q gives a relation between x and y, which is one equation of the complete solution. This equation may enable us to eliminate x or y from one of the other equations in (2), and then another integral relation may be found; this will be the second equation of the solution. Since, by a well-known principle of algebra, the equal fractions d, d, dz, are also equal to P' ' R 132 DIFFERENTIAL EQUATIONS. [CH. XI. 1dx + m dy +ndz l'dx + m'dy + n'dz et IP + mQ nR ' 'P + m'Q + n'R the system of equations dx_ dy dz l dx + n dy + - n dlz _ l'dxc m'dy + n'dz P -Q= R 1P+iQ+ nR - 'P+m'Q+nlR' (3) are all satisfied by the same relations between x, y, z, that satisfy (2). It may be possible by a proper choice of multipliers, 1, mn, n, 1', m', n', etc., to obtain equations which are easily solved, and whose solutions are the solutions of (2). In particular, 1, nM, n, may be found such that 1P + Q + nR O, (4) and consequently l dx -- m ndy + n dz - 0. (5) If I dx + m dy + n dz be an exact differential, say du, then u = a is one equation of the complete solution. If ', m', n', can be chosen so that l'P + m'Q + n'R = 0, and l'dx + m'dy + n'dz is at the same time an exact differential, clv, then, since dv is also equal to zero, v b is the second equation of the complete solution. The two component solutions must be independent. Ex. 1. Solve the simultaneous equations x2 x dy =dz X_- y2 _- 2 2 xy 2xz The equation formed by the last two fractions reduces to dy_ dz y z which has for its solution y - az. Using x, y, z, as multipliers like 1, m, n, above, dx _ dy _ dz _xdx + ydy + zdz X2 - y2 _ 2 2 Xy 2 X X(2 + +y2 + 2) ~ iOO.] INTEGRALS OF SIMULTANEOUS EQUATIONS. 133 The equation formed by the last two fractions has for its solution X2 + y2 + z2 = bz. The complete solution consists of these two independent solutions. Ex. 2. Solve dx _ dy dz mz - ny nx - Iz ly - mx By using the multipliers 1, m, n, one gets the equal fraction dx + mdy + n dz therefore I dx + m dy n dz = 0; whence Ix + my + nz = cl. The multipliers x, y, z, used in a similar manner, give xx d+ y dy + z dz = 0, whence x2 + y2 + zZ2 = k2. These two integrals forln the complete integral of the set of equations. Ex. 3. Solve xdx=dy_=d y2z xz y2 Ex. 4. Solve d=dy= dz x2 y2 nxy Ex. 5. Solve dx _ dy dz y2 x2 x2y2z2 Ex. 6. Solve adx _ bdy = cdz (b - c)yz (c - a)zx (a - b)xy 100. The general expression for the integrals of simultaneous equations of the first order. If the first member, 1 dx + m dy + n dz, of (5) Art. 99 be an exact differential, du, then, since au au u du dx + - cdy + dz, ax ay az 1, m, n, are proportional to au au a, respectively; and therefore, (4) Art. 99 may be written 134 DIFFERENTIAL EQUATIONS. [CH. XI. (1) Pau + Q at + R a_ o= ( Dx dy ôz Hence, if u = a be one of the integral equations of the system (2) Art. 99, then u = a also satisfies (1). Conversely, if u = a be an integral of (1), it is also an integral of the system (2) Art. 99. For, since the denoininator of du, Du D7 u c dx +- dy + - dz ôx ôy Dz P + Q +Ra Dx ay az which is formed by means of c = d- - and the multipliers Du Du Du ~ Q R Dx' Dy" Dz' is thus 0, the numerator also must equal zero. ax ay z But the numerator is the total differential of u, and hence u = a is an integral of the system (2). Therefore, in order that u = ca may be an integral of (1), it is necessary and sufficient that u = a be an integral of the system (2) Art. 99, and conversely. Moreover, any function whatever of the u and the v of Art. 99 is also a solution of (1); for example, < (uC, v) = < (a, b) = r, or V (u, ) = 0, which is equally general, since c can be involved in the arbitrary function. This can be verified directly. Hence, if u = a, v = b, be independent integrals of the system (2) Art. 99, < (u, v) = o is the general expression for the integrals of these equations. The arbitrary functional relation may just as well be written in the form u =f(v). This deduction will be used in Art. 115. 101. Geometrical meaning of simultaneous differential equations of the first order and the first degree involving three variables, ~ 101.] GEOMETRICAL MEANING. 135 Equations (1) or (2) Art. 99 will determine, for each point (x, y, z), definite values of dx and d-; that is, these differential equations determine a particular direction at each point in space. Therefore, if a point moves, so that at any moment the co-ordinates of its position and the direction cosines of its line of motion (these cosines being proportional to dx, dy, dz, and hence to P, Q, R, by (2) Art. 99) satisfy the differential equations, then this point must pass through each position in a particular direction. Suppose that a moving point P starts at any point and moves in the direction determined for this point by the differential equations to a second point at an infinitesimal distance; thence, under the same conditions to a third point; thence to a fourth point, and so on; then P will describe a curve in space, whose direction at any one of its points and the co-ordinates of this point will satisfy the given differential equations. If P start from another point, not on the last curve, it will describe another curve; through every point of space there will thus pass a definite curve, whose equation satisfies the given differential equations. These curves are the intersections of the two surfaces which are represented by the two equations forming the solutions; for, these two equations together determine the points and the ratios of dx, dy, dz, thereat which satisfy the differential equations. Moreover, the curves are doubly infinite in number; for they are the intersections of the surfaces represented by the independent integrals u = a, v- b, and each of these equations contains an arbitrary constant which can take an infinite number of values. Thus, the locus of the points that satisfy the differential equations of Ex. 1, Art. 99, is the curves, doubly infinite in number, which are the intersections of the system of planes whose equation is y= az, with the system of spheres whose equation is x2 + y, + z- = bz; 136 DIFFERENTIAL EQUATIONS. [CH. XI. and the locus of the points that satisfy the equations of Ex. 2, Art. 99, is the curves which are the intersections of all the planes represented by lx + my + nz = c, c having an infinite number of values, with all the spheres x2+ y2 + z2 k2, k having an infinite number of values. 102. Single differential equations that are integrable. Condition of integrability. The equation Pdx + Qdy + Rdz= (1) has an integral u = a, (2) when there is a function u whose total differential du is equal to the first member of (1), or to that member multiplied by a factor. If (1) have an integral (2), then, since O u O + Ou Ou ldu= -c dx +- dy + - dz, Ox ôy az P, Q, R, must be proportional to -, au, j; that is, ax ay ôz Ox ôy uO IL-I O Oz These three conditions can be reduced to one involving the coefficients P, Q, R, and their derivatives. On differentiating the first of these three equations with respect to y and z, the second with respect to z and x, and the third with respect to x and y, there results, ~ 102, 103.] SINGLE INTEGRABLE EQUATION. 137 p1_ / P &2u -RL + Q / P- + dP 2R Q + d a y ax ay y x ax Q a Oi +, Q -a Q_ 32 n _ R aO + O IR. 0 z ô ôy yz ay ay L + DR ô% U ÔP Ol OR O = __ OP Rx Ox z+ x dz + whence, on rearranging, cornes aP_ aQ\1 a,~ pa, 1 (?_ =x RQ -x -y ( Q _P _ ] _ i Qz ôy) Ry dz Q (OR I \ax az aOz ax On multiplying the first of the last three equations by R, the second by P, the third by Q, and adding, there is obtained, p Q _R) Q (OR _P) P R dQ)o~ PfaQ Q _^ _^ o, (3) Dz dy ôdx ôz ôy dôz () the relation that must exist between the coefficients of (1) when it has an integral (2). Conversely, when relation (3) is satisfied, equation (1) has an integral;* and hence (3) is the necessary and sufficient condition that (1) be integrable. It is called the condition or criterion of integrability of the single differential equation (1); and is easily remembered, for P, Q, R, x, y, z, appear in it in a regular cyclical order. 103. Method of finding the solution of the single integrable equation. Suppose that the condition for the integrability of Pdx + Qdy + Rdz = (1) For proof of this, see Note H. 138 DIFFERENTIAL EQUATIONS. [CH. XI. is satisfied; a nmethod has now to be devised for finding its solution. Pdx can only come from the ternis of the integral that contain x, Qdy from the terms that contain y, and Rdz from the terms that contain z. Hence the integral of (1) is found in the following way: Consider any one of the variables, say z, as constant, that is, take dz = 0, anc integrate the equation Pdx + Qdy = 0. (2) Put the arbitrary constant of integration that inust appear in the integral of (2) equal to an arbitrary function of z. This is allowable because the arbitrary constant in the integra] of (2) is a constant only with respect to x and y. On differentiating the integral just found, with respect to x, y, and z, andcl coinparing the result with (1), it will be possible to determine the constant appearing in the integral of (2) as a particular function of z. Equations which are homnogeneoi in x, y, z, like those in Art. 9 in x, y, are always integrable. The initial step in solving these equations is the substituti i of zu for x, and of zv for y.* NOTE. That an equation of the fori Pdx + Qdy + Rdz + Tdt +. =0, involving more than three variables, may have an integral, condition (3) Art. 102 must hold for the coefficients of all the terms taken by threes. All the conditions thus formed are, however, not independent.t Ex. 1. Solve (y + z)dx + (z + x)dy + (x + y)dz = 0. Here, the condition of integrability is satisfied. * See Johnson, Diferential Equations, Art. 250. t See Johnson, Differential Equations, Arts. 252-254; Forsyth, Differential Equations, Arts. 163, 164. For a complete proof of these propositions, see Forsyth, Theory bf Differential Equations, Part I., pp. 4-12. ~ 103.] EXAMPLES. 139 Suppose that z is a constant, then dz = 0, and the equation becomes (y + z)dx +(z + x)dy = 0; and this on integration yields (y + z)(z + x) = (z). Differentiation with respect to x, y, z, gives (y + z)dx +(z + x)dy +(x + y)dz + 2zd -dddz =0. dz Comparison with the given equation shows that 2zdz - do = 0; whence (z) = z2 + c2. Therefore, (y + z)(z + x) = 2 + c2; or, reducing, xy + yz + zx = c2 is a solution of the given equation. This example can be solved more easily by rearranging the terms in the following way: x dy + y dx + y dz + z dy + z dx + x dz = 0, where the integral is seen at a glance to be xy + yz + zx = c2. It is well to try to obtain the integral by rearranging the terms, before having recourse to the regular method. Ex. 2. Solve xdx+ydy+zdz+ zd-xd 3ax2dx2bydy+ (x2+ y2+ z2) 2 + z2 Here there is no need to apply the condition of integrability, for the several parts are obviously exact differentials; the integral is immediately seen to be /x2 + y2 + z' + tan-' x+ ax3 + by2 + cz = k. Ex. 3. Solve (y + z)dx + dy + dz = 0. Ex. 4. Solve zydx = zxdy + y2dz. Ex. 5. Solve (22 +2xy+2xz2+ 1)d+ dy + 2zd = 0. Ex. 6. Solve (y2 + yz)dx + (xz + z2)dy + (y2 - xy)dz = 0. 140 DIFFERENTIAL EQUATIONS. [CH. XI. 104. Geometrical meaning of the single differential equation which is integrable. Suppose that the equation Pdx + Qdy + Rdz = 0 (1) satisfies the condition of integrability, and that its solution is F(x, y, ) = C. (2) Equation (2) represents a single infinity of surfaces, there being one arbitrary constant. This constant can be determined so that (2) will represent the surface which passes through any given point of space. If a point is moving upon this surface in any direction, the co-ordinates of its position and the direction cosines of its path at any moment, which are proportional to dx, dy, dz, satisfy (1), since (2) is the integral of (1). Also for each point (x, y, z) there will be an infinite number of values of d and dd which will satisfy (1); therefore, a point that is dz dz moving in such a way that its co-ordinates and the direction cosines of its path always satisfy (1) can pass through any point in an infinity of directions. But, when passing through any point, it must reluain on the particular surface represented by the integral (2) which passes through the point; hence all the possible curves, infinite in number, which it can describe through that point must lie on that surface. It has been shown in Art. 101 that a point which is moving subject to the restrictions imposed by tle two equations (1) Art. 99 can describe only one curve through any one point; on the other hand, a point that is moving subject to the restriction of a single integrable equation can describe an infinity of curves through that point; all the latter curves, however, lie upon the same surface. For example, a point passing through the point (1, 2,3) in such a direction as to satisfy the equations of Ex. 1, Art. 99, must move along the intersection of the plane having the equation 3y=2z and the sphere whose equation is ~ 104, 105.] GEOM3ETRICAL MEANING. 141 3 (x2 + y' + -z) = 14z. A point moving so as to satisfy the equation of Ex. 1, Art. 103, can pass through (1, 2, 3) in an infinity of directions, but all these possible paths lie upon the surface having the equation xy + yz + zx- 11. 105. The locus of Pdx + Qdy + Rdz = O is orthogonal to the locus of dx = dy _ dz The equation P Q R Pdx + Qdy + Rdz 0 (1) means, geometrically, that a straight line whose direction cosines are proportional to dx, dy, dz, is perpendicular to a line whose direction cosines are proportional to P, Q, R.* Therefore, a point that is moving subject to the condition expressed by (1) must go in a direction at right angles to a line whose direction cosines are proportional to P, Q, R. On the other hand, the equations dx dy _ dz (2) P Q R mean, geometrically, that a straight line whose direction cosines are proportional to dx, dy, dz, is parallel to a line whose direction cosines are proportional to P, Q, R. Therefore, a point that is moving subject to the conditions expressed by (2) must go in a direction parallel to a line whose direction cosines are proportional to P, Q, R. Therefore, the curves traced out by points that are moving' subject to the condition (1) are orthogonal to the curves traced out by points that are moving subject to the conditions (2). The former curves are any of the curves upon the surfaces represented by (1); therefore the curves represented by (2) are normal to the surfaces represented by (1). If (1) be not integrable, there * C. Smith, Solid Geometry, Art. 24. 142 DIFFERENTIAL EQUATIONS. [CH. XI. is no family of surfaces which is orthogonal to all the lines that form the locus of equations (2). The principle deduced in this article will be employed in Art. 118 of the next chapter. 106. The single differential equation which is non-integrable. When the condition of integrability is not satisfied for Pdx + Qdy + Rdz = 0, (1) there is no single relation between x y, z, as, for example, f(x, y, z) = c, that will satisfy (1). If, however, there be assumed some integral relation, 4 (, y, ) = a, (2) which on differentiation gives the differential relation a dx + ady + dz = 0,. (3) ôx ay dz two integral relations can be found which together satisfy (1) and (3), this being the case discussed in Art. 99. Of course, (2) is one of these relations. Suppose that F(x, y, z) = b (4) is a relation which with (2) forms the complete solution of equations (1) and (3). In Art. 101 it was shown that the locus of the complete solution of (1) and (3) consists of the curves of intersection of (2) and (4); hence, geometrically, this solution of (1) amounts to finding the curves satisfying (1) that lie on the surfaces represented by (2). Ex. The equation (1) xdx + y dy+ c 1 - - -dz = a2 b2 is one for which the condition of integrability is not satisfied. Suppose that the relation x2 y2+ Z2= (2) asuml be a2 assumed. be assumed. ~ 106.] EXAMPLES. 143 In virtue of (2), (1) may be written in the form (3) xdx + y dy zdz = 0; whence (4) x2 + y2 + z2 c2. Thus (2) with (4) gives a solution of (1). Had a relation other than (2) been assumed, a co-relation other than (4) would have been obtained. The geometric interpretation is, that the lines upon the ellipsoid represented by (2) which satisfy (1), have been determined; and have been found to be the intersections of the family of spheres whose equation is (4) with that ellipsoid. EXAMPLES ON CHAPTER XI. 1. d+2 2 2 x+2=3e 3. x + 2 y- e 2x 3 dx + dtJ + 2 x + y-= 4 y - =2x dt dt tdx2 dx dx -*- + 2 x d y 4 e2t__2 " 4dz 3 0 32d+4dZ-3='0. dlt (it dx dx lx cldx 2. 4d-G+9 d+2x+31y=et 4. -+4x+3y=t dt dt dt 3 dx _ 7 y + x + 24 y =3. dy 2 x + 5 y =et. dt dt dt 5. tdx =(t-2x)dt t dy = (tx + ty + 2x - t)dt. 6. x2dx2 + y2dy2 - z2dz2 + 2 xy dx dy = 0. 7. (x2y - y3 _ y2z)dx + (xy2 - x2z - x)dy + (xy2 + x2y)dz = 0. 8. (y2 + yz + z2)dx 4- (X2 + xz + z2)dy + (X2 + xy + y2)dz = 0. 9. (yz + xyz)dx + (zx + xyz)dy + (xy + xyz)dz = 0. 10. z(y + z)dx + z(u - x)dy + y(x - u)dz + y(y + z)du = 0. 11. (2 x + y2 + 2 xz)dx + 2 xydcy + x2dz = du. 12. z dz + (x - a)dx = {h2 - z2 - (x - a)2}) dy. 13. + 4x + y =te3t 14. dx + 2y= dt2 dt2 d2y + y - 2 =os2t. d 2 = 0. dt2 dt2 144 DIFFERENTIAL EQUATIONS. [CH. XI. 15. dx _ cl2J _ ci~x tvliei~ 15. dx - mz 16. dx =c dy= where dt X YZ dy= _z - nx X= ax + by + cz + d dt Y = a'x + b'y + c'z + dl = mx - ly. dt Z = altx + b"y + ctz + dI. 17. Show that the integrals of the system dx_= ax + by + c, dy= afx+ bfy + c, dt dt are (a + ma') (x + nly) + c + m1c' = Ae(a+la')t, (a + m2a') (x + m2y) + c + m2c' = A2e(a+m2a)t, where mi, m2, are the roots of a'm2 + (a- bt)m - b = 0; and obtain a similar solution for the system d'2x d'2y d = ax -- by, dy = ax + b'y. dt2 dt2 (Johnson, Dif. Eq., Ex. 16, p. 269.) 18. Find the equation of the path described by a particle subject only to the action of gravity, after being projected with an initial velocity v0 in a direction inclined at an angle ( to the horizon. 19. Determine the path of a projectile in a resisting medium such as air when the retardation is c times the velocity, given that the initial velocity is v0 in a direction inclined at an angle P to the horizon. 20. Find the path described by a particle acted upon by a central force, the force being directly proportional to the distance of the particle. 21. The two fundamental equations of the simple analytical theory of the transformer are Ri i + 1 " - - el, dt dt dcli di 0 R2i2 + L2 -t +M -- O, dt dt where il, i2, denote the currents, Ri, R2, the resistances, L1, L2, the coefficients of self-induction of the primary and secondary currents respectively, e1 the impressed primary electromotive force, and M the mutual induction. ~ 106.] EXAMPLES. 145 Show that, e, il, i2, and t being variable, the differential equations for the primary and secondary currents respectively are, d2~i1 dil de1 (L1L2 - X12) d + (L1R2 + L2R1) 3d + RlR2il = R2el + L2-, dt2 dt dt (LiL22 - 12)!t + (L1R2 + L2Ri) di2 + R1R2i2 = - M. dc?2 dt dt (Bedell, The Principles of the Transformer, Chap. VI.) 22. The general equations for electromotive forces in the two circuits of a transformer with capacities cl and c2 being idt Lii di2 e = f(t) = i+ lc + L1i d + 1ddi2, Ci dt cldt 0 *= + R2C2 + L2 -+ + i2 C2 dit dt where e, il, is, t, are variable, show that the differential equations for the primary and secondary currents are (L2l2 -12) dt+(211L2 + R2L2i)S!+ f + -+ R1R2) 1 \C2 C dt C1C2 2 C (L1L2 - i2) + ^i +, + i + i 2 ct 2c dij ctt ci C dt2 + iRi+ `2 I i2 L1 (- (2 ---! - (1?1L 2 -i- -- d f t k t2. \ C i Cl/dt C1C2 (Bedell, The Principles of the Transformer, Chap. XI.) L 146 DIFFERENTIAL EQUA1TIONS. [CH. XII. CHAPTER XII. PARTIAL DIFFERENTIAL EQUATIONS. 107. Definitions. Partial differential equations are those which contain one or more partial derivatives, and niust, therefore, be concerned with at least two independent variables.* The derivation of partial differential equations will be discussed in Arts. 108, 109; equations of the first order will be considered in Arts. 110-123; and those of the second and higher orders in the remaining part of the chapter. These equations, excepting the ones treated in Arts. 117, 134-136, will involve only three variables. In what follows, x and y will usually be taken as the independent variables, and z as dependent; the partial differential coefficients -, z, will be denoted by p and q respectively. 108. Derivation of a partial differential equation by the elimination of constants. Partial differential equations can be derived in two ways: (a) by the elimination of arbitrary constants from a relation between x, y, z, and (b) by the elimination of arbitrary functions of these variables. To illustrate (a) take (, (, z, a, b) = (1) a relation between x, y, y, the latter variable being dependent upon x and y. In order to eliminate the two constants a, b, * Equations with partial derivatives were at first studied by DI'Alembert (see p. 173), and Euler (see p. 64), in connection with problems of physics. ~ 107, 108.] DERIVATION. 147 two more equations are required. These equations can be obtained from (1) by differentiation with respect to x and y; they will be ax ôz ay az By means of these three equations, a and b can be eliminated, and there will appear a relation of the form F(x, y, z,, q) =, (2) a partial differential equation of the first order. In (1) the number of constants eliminated is just equal to the number of independent variables, and an equation of the first order arises. If the number of constants to be eliminated is greater than the number of independent variables, equations of the second and higher orders will, in general, be derived. The following examples will illustrate this. In these examples, z is to be taken as the dependent variable. Ex. 1. Form a partial differential equation by the elimination of the constants h and k from (x - h)2 + (y - k)2 + z2 = c2 Differentiating with respect to x and y, x - h + zp = 0, y - k + zq = 0. Substituting the values of x - h, y - k from the last two equations in the given equation, z2(p2 + q2 + 1)= c2. Ex. 2. Form the partial differential equation corresponding to z = ax + by + ab. Ex. 3. Eliminate a and b from z = a(x + y) + b. Ex. 4. Eliminate a and b from z = ax + a2y2 + b. Ex. 5. Eliminate a and b from z = (x + a)(y + b). Ex. 6. Form a partial differential equation by eliminating a, b, c from x2+- 2+ =.2 2 &2 c2 148 DIFFERENTIAL EQUATIONS. [CH. XII. 109. Derivation of a partial differential equation by the elimination of an arbitrary function. To illustrate (b) of Art. 108, suppose that u and v are functions of x, y, z, and that there is a relation between u and v of the form (, v) =, (1) where b is arbitrary. The relation may also be expressed in the form u =f(v), where f is arbitrary. It is now to be shown, that, on the elimination of the arbitrary function p from (1), a partial differential equation will be formed; and, moreover, that this equation will be linear, that is, it will be of the first degree in p and q. Differentiation of (1) with respect to each of the independent variables x and y gives + ( ' + L 23I ( +pX 0, ôa\ x az aav ax dz O u + qa + a(av + qa =0. \ay z j dav dy aj) Elimination of ', -', from these two equations results in Oau iv c a + P, +. dv) (+u + q 9U 9V+, dv\) O Oz dy Ozz x dz and this can be rearranged in the form Pp+Qq=R, (2) aU av ait av where P= v ay dz dz dy R au av aOu a0v Rou xo O yO ax ay dy ax Thus, from (1), which involves an arbitrary function ~, a partial differential equation (2) has been obtained, which does not contain q and is linear in p and q. ~ 109, 110.] INTEGRALS OF NON-LINEAR EQUATIONS. 149 When the given relation between x, y, z contains two arbitrary functions, the partial differential equation derived therefrom will, except in particular cases, involve partial derivatives of an order higher than the second.* Ex. 1. Eliminate the arbitrary function from z = e'lY (x - y). Differentiating with respect to x, p = e"I(x - y). Differentiating with respect to y, q = ne'lY (x - y) - eny' (x - y); and, therefore, q = nz -, that is, p + q = nz. Ex. 2. Form a partial differential equation by eliminating the arbitrary function from z = F(x2 + y2). Ex. 3. Eliminate the function 0 froIn lx + my + nz = 0(x2 + y2 + z2). Ex. 4. Eliminate the function front z = y2 + 2f(- + logy ) Ex. 5. Eliminate the arbitrary functions f and 0 from =f (x + ay)+ 0(x - ay). PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER. 110. The integrals of the non-linear equation: the complete and particular integrals. In Art. 108 it was shown how the partial differential equation F(, y,, p, q) =0 (1) may be derived from / (x, y, a, b)= 0. (2) Suppose, now, that (2) has been derived from (1), by one of the methods hereafter shown; then the solution (2), which lias as many arbitrary constants as there are independent variables, is called the complete integral of (1). A particular integral of (1) is obtained by giving particular values to a and b in (2). * See Edwards, Differential Calculus, Arts. 509-514; Williamson, Differential Calculus, Arts. 315-319; Johnson, Differential Equations, Arts. 299-301. 150 DIFFERENTIAL EQUATIONS. [CH. XII. 111. The singular integral. The locus of all the points whose co-ordinates with the corresponding values of p and q satisfy (1) Art. 110, is the doubly infinite system of surfaces represented by (2). The system is doubly infinite, because there are two constants, a and b, each of which can take an infinite nuniber of values. Since the envelope of all the surfaces represented by < (x, y, z, a, b) 0 is touched at each of its points by some one of these surfaces, the co-ordinates of any point on the envelope with the p and the q belonging to the envelope at that point must satisfy (1); and, therefore, the equation of the envelope is an integral of (1). The equation of the envelope of the surfaces represented by (2) is obtained in the following way: * Eliminate a and b between the three equations, < (x, y,, a, b) = 0, = O, dcda db and the relation thus found between x, y, z is the equation of the envelope. This relation is called the singular integral; it differs from a particular integral in that it is not contained in the complete integral; that is, it is not obtained from the complete integral by giving particular values to the constants. (Compare Arts. 32, 33.) 112: The general integral. Suppose that in (2) Art. 110, on( of the constants is a function of the other, say b =-f(a), ther this equation becomes (x v, z, z,,f (a)) = o, (1; which represents one of the families of surfaces included iI the system represented by (2). The equation of the envelop{ * For proof see C. Smith, Solid Geometry, Arts. 211-215; W. S Aldis, Solid Geometry, Chap. X. ~ 111, 112.] THE GENERAL INTEGRAL. 151 of the family of surfaces represented by (1) will also satisfy (1) Art. 110, for reasons similar to those given in the case of the singular integral. Moreover, this equation will be different from that of the envelope of all the surfaces, and it is not a particular integral. It is called the general integral; and it is found by eliminating a between (x, y(,, f(a)) = 0, and d= 0. da These two equations together represent a curve, namely, the curve of intersection of two consecutive surfaces of the system 4( (x, y, z, a, f(a)) = 0. The envelope of the family of surfaces, being the locus of the ultimate intersections of the surfaces belonging to the family, that is, of the intersections of consecutive surfaces, contains this curve to which the name characteristic of thlr envelope has been given. Hence the general integral may be clefined as the locus of the characteristics. Other relations may appear in the process of deriving the singular and the general integrals from the complete integral, but it is beyond the scope of this work to discuss such relations. When one has performed the operations necessary to find the singular and the general integrals, he should test his result by trying whether it satisfies the differential equation. (Compare Arts. 33-38.) In the case of every equation, the general integral and the singular integral, as well as the complete integral, must be indicated or the equation is not considered to be fully solved. The complete integral is to be found first, and from it the other two are to be derived.* It is evident that the locus of the singular integral will be the envelope of the loci of all the other integrals, of the general as well as of the complete. * The distinction between the three kinds of integrals of partial differential equations was made by Lagrange in Jlemoirs of the Berlin Academy, 1772, 1774. 152 DIFFERENTIAL EQUATIONS. [CI-. XII. Ex. In Ex. 1, Art. 108, the differential equation z2( p2 + q2 + 1)= 2 (1) was derived from ( - h)2 +(yk)2+ 2 = c2. (2) The latter equation, which contains two arbitrary constants, is the complete integral of the former; it represents the doubly infinite system of spheres of radius c, whose centres are in the xy plane. A particular integral of (1) is obtained by giving h and k particular values in (2); thus, (x - 2)2 + (y - 3)2 + z2 = c2 is a particular integral. The singular integral of (1) is the equation that represents the envelope of these spheres; it is obtained by eliminating h and k from (2) by means of the relations derived by differentiating (2) with respect to h and k. The differentiation gives x - h = 0, y - k 0; on substituting these values in (2), h and k are eliminated, and there results the equation z {= c. This satisfies equation (1), and, therefore, is the singular integral. It represents the two planes that are touched by all the spheres represented by (2). Suppose, now, that one of the constants is made a function of the other, say, that k = h. Then the centres, since their co-ordinates have that relation, are restricted to the straight line y = x in the xy plane; and of the system of spheres representing (2) there will be chosen a particular family, namely, (x - h)2 + (y - h)2 + z2 = c2. (3) The envelope of this family is the tubular surface, in this case a cylinder, which is generated by a sphere of radius c, when its centre moves along the line y = x. The equation of this envelope is a general integral; it is found by eliminating h from (3) by means of the relation obtained by differentiating (3) with respect to h. The differentiation gives x - h + y - h = 0, whence h =- ~(x + y). ~ 113.] INTEGRAL OF THE LINEAR EQUATION. 153 Substituting this value of h ill (3), x2 + y2 - 2xy + 2 z2 = 2 c2, which is a general integral. If the relation between the constants were assumed to be k2 = 4 ah, the corresponding general integral would be the equation of the tubular surface generated by a sphere of radius c, whose centre moves along the parabola y2 = 4 ax in the xy plane. 113. The integral of the linear equation.* In Art. 109 it was shown that from an arbitrary functional relation - 0 (u, V) O (1) there is derived, by the elimination of the function %, a linear partial differential equation Pp + Qq = R. (2) Suppose that (1) has been derived from (2); then 4(zu, v)= O is called the general solution of (2). Since 4( is an arbitrary function, the solution (1) is more general than another solution of (2) that merely contains arbitrary constants. For instance, Ex. 2, Art. 109, shows that the general solution of yp - xy = O is z = F(x2 + y22), where F denotes an arbitrary function. The arbitrary function F may take various forins, as, z = Ct (2 + y2)2+ b(2 + y2), z = a sin (x2 + y2) + b, etc., which are all solutions of the differential equation, and are included in the general solution above. * The student will find it of great advantage to read C. Smith, Solid Geometry, Arts. 216-226; W. S. Aldis, Solid Geometry, Arts. 142-151, in connection with this and following articles. 154 DIFFERENTIAL EQUATIONS. [CH. XII. 114. Equation equivalent to the linear equation. Thie type of a partial differential equation which is linear in p and q is Pp+ Qq=R, (1) P, Q, R being functions of x, y, z. Suppose that u = a is any relation that satisfies (1); differentiation with respect to x and y gives u + u Ox Oz -~+ au=0; Oy Oz Ou Ou Ox Oy whence p = —, q ôz Oz Substitution of these values of p and q in (1) changes it to p u+Q +Rtau. (2)u dôx dy dz Therefore, if 'u = a be an integral of (1), u = a also satisfies (2). Conversely, if u = a be an integral of (2), it is also an integral of (1). This can be seen by dividing by a and substiOz tuting p and q for the values above. Therefore equation (2) can be taken as equivalent to equation (1). 115. Lagrange's solution of the linear equation. In Art. 100 it was shown that p(u, v) = O is a general integral of (2) Art. 114 when u = a, v = b are independent integrals of the systern of equations dx _ dy _ dz P Q R ~ 114-116.] LAGRANGE'S SOLUTION. 155 Hence the following rule may be given: To obtain an integral of the linear equation of the form Pp + Qg = R, find two independent integrals of dx _ dy dz P Q R' let them be u = a and v = b; then 4(u, v)= 0, where q is an arbitrary function, is an integral of the partial differential equation. Instead of U(zu, v) 0, there can with equal generality be written u =f(v), where f denotes an arbitrary function. This is known as Lagrange's solution of the linear equation; the auxiliary equations (3) are called Lagrange's equations; and the curves of intersection of the surfaces represented by the integrals of (3) are called Lagrangean lines. 116. Verification of Lagrange's solution. The truth of Lagrange's solution may also be shown in the following way. Form the differential equations corresponding to u = a and v = b, by eliminating the arbitrary constants a and b; this gives 9u ac au Ou dx + u dz = O, dx dy az v dx+ v dy +v dz =, ax ay az * Joseph Louis Lagrange (1736-1813) was one of the greatest mnathematicians that the world has ever seen. He wrote much on differential equations, and the theory of the linear partial equation was first given by him. He discussed the case of three variables and gave the solution in a memoir in the Berlin Academy of Sciences in 1772; he treated singular solutions in a memoir of 1774; and in memoirs of 1779 and 1785 he gave a generalised method applicable to equations having any number of variables. See footnote, page 40. 156 DIFFERENTIAL EQUATIONS. [CH. XII. dx dy dz whence du v du aDv -u Dv Du dv du av Ou av az y -z x xz a y y y ox But, in Art. 109, it was found that the equation derived from q (u, v) = 0 by eliiniating < is a(u av au dv\ ( fu av au av _ u dv du av D\y Dz Dz dq kDz Dx ax Dz ) Qx ay ay ax Comparison shows that these equations have the forms dx _ dy _ dz P Q R and Pp + Qq = R, respectively. Ex. 1. Solve xzp + yzq - xy. Dividing by xyz, +; forming the auxiliary equations, y dx x dy z dz. Integrating the equation formed by the first two terms, Y = C. Also ydx + xdy= 2 zdz; whence z2 - xy c. Therefore, the solution is z2 - xy - (-), or f (2 -xy =0. Ex. 2. Solve p + q = a Ex. 3. Solve (mz - ny)p + (nx - lz)q = ly - mx. Ex. 4. Solve x2p + y2q = z2. y2zp Ex. 5. Solve y2- + xzq y2. 117. The linear equation involving more than two independen variables. If there be n functions u1, u2,.., iO of n + 1 varia bles z, x1x, x2..., x z being dependent and the other variable ~ 117.] MORE THAN TWO INDEPEINDENT lVALIABLES. 157 independent, then the arbitrary function t+ can be eliminated from 4( (u,, 2,..., z)= O (1) by an extension of the method used in Art. 109. The result will be a linear partial differential equation of the form dx dx2 dx. a, + P, + '"~ + '~~ -x ~ ~1~. (2) Moreover, on forming the differential equations corresponding to Ui -= c, u2 = c, *.., u, c,, by eliminating the constants c", c2., c,, and proceeding as in Art. 116, there will be obtained dx_ _ dx. _ dx,_ dz 11 P2 P' R Hence the following rule may be given: In order to deduce the general integral of the partial differential equation (2), write down the auxiliary equations (3), and find n independent integrals of this systein of equations; let these integrals be Ul - Cl, U2 - C2, '** ' tn C=n tlen >(u1, 2,' *u un) = 0 where 4> denotes an arbitrary function is the integral of the given. equation. Suppose that u = c is an integral of (2); then Du ôz dx. since _ —, (i=_,2,.. n), axi - a uDz equation (2) can take the equivalent form Duv u Du 0 Pl + + *~ *- +n + R. (4) alx aDx2 Dan Da 158 DIFFERENTIAL EQUATIONS. [CH. XII. Ot Ot 3t Ex. 1. Solve (t+y+z)-t +(t+x+z)- t +(t+x+y) =x+y+z. ax ay az The auxiliary equations are dt dx dy dz x+y+z y+z+t z+t+x t+x+y' whence, dt + dx + dy + dz _ dt - dx 3(t + x + y + z) x-t i Cl from this, log (t + x + y + z) = log hence, (x - t)(t + x + y + z)3 = C1; similarly, (y - t)(t + x + y + z)~ = c2, and (z - t) (t + x + y + z)3 = C3. Hence the solution is {(x - t), (y - t)u, ( - t)u}= O, where u =(t + x + z)3. Ex. 2. Solve x - + y - s + z - xyz. Ox Od az 118. Geometrical meaning of the linear partial differential equation. In Art. 105 it was shown that the curves whose equations are integrals of dx _ dy dz (1) are at right angles to the system of surfaces whose equation satisfies Pcdx + Qdy +- Rdz 0O. (2) Suppose that - a, v = b are any pair of independent integrals of (1). Let a take a particular value, say c. The surface represented by u = ca is intersected by the system of surfaces whose equation is v = b, in an infinite number of curves, a curve for each one of the infinite number of values that b can have. Thus u = a, repre ~ 118, 119.] SPECIAL METFIODS OF SOLUTION. 159 sents a locus which passes through, or upon which lie, curves infinite in number, that are orthogonal to the surfaces represented by (2). Therefore, since the general integral of Pp + Qq = R (3) is an arbitrary function of integrals of equations (1), any integral of (3) passes through a system of lines that are orthogonal to the surfaces forming the locus of (2); and hence the surfaces represented by (3) are orthogonal to the surfaces represented by (2). * 119. Special methods of solution applicable to certain standard forms. There are a few standard forms to which many equations are reducible, and which can be integrated by methods that are sometimes shorter than the general method wlich will be shown in Art. 123. These forms will now be discussed. Standard I. To this standard belong equations that involve p and q only; they have the form F(p, q) =. (1) A solution of this is evidently z = ax + by + c, if a and b be such that F(a, b) = 0; that is, solving the last equation for b, if b =f(a). The complete integral then is =ax + yf (a) + c. (2) The general integral is obtained by putting c = 4 (a), where c denotes an arbitrary function, and eliminating a between z = ax + yf(a) + '( (t), and 0 = x + yf'(a) + -'(a). * Arts. 119-122 closely follow Forsyth, Diferential Equations, Arts. 191-196. 160 DIFFERENTIAL EQUATIONS. [Ci. XII. The singular integral is obtained by eliminating c aand c between the complete integral (2) and the equations formed by differentiating (2) with respect to a and c; that is, between z = ax + yf(a) + c, 0 = x + yf' (a), 0 =1; the last equation shows that there is no singular integral. Ex. 1. Solve (1) p2 + q2 = 2. The solution is z = axr + by + c, if a2 + b2 = m2. Therefore, the complete solution is (2) z = ax + x/m2 - a2y + c. To find the general integral, put c = f(a); then z = ax + -/m- - a2 y + f(a); differentiate with respect to a, 0=x- a y +ft(a); O,/i2 - a2 and eliminate a by means of these two equations. A developable surface is the envelope of a plane whose equation contains only one variable parameter.* Therefore, the general integral in this case represents a developable surface. In particular, if c or f(a) be chosen equal to zero, then the result obtained by eliminating a is (3) z2 - m2(x + y2). The complete integral (2) represents a doubly infinite system of planes; the particular integral obtained by putting c equal to zero represents a singly infinite system of planes passing through the origin; and the general integral (3) represents the cone which is the envelope of the latter system of planes. Ex. 2. Solve (1) x2p2 + y2q2 = z2. This may be written zxaz +2 — Yd) 1. Put d =dX, =d, \zxi \Zay} X y dz=dZ; whence X=logx, Y=logy, Z = logz; the equation then z * See C. Smith, Solid Geometry, Art. 221. ~ 120.] STANDARD II. 161 becomes a) ( Z)1, which comes under Standard I. From the preceding example the complete integral is Z = aX+ /1 - aY + logc; hence, z = cxayl- a2, which is the complete integral of (1). The singular integral is z = 0; the general integral is to be found in the usual way. Ex. 3. Solve 3p2 -2 q2 = 4pq. _ Ex. 4. Solve q = e. Ex. 5. Solvepq = k. 120. Standard II. To this standard belong equations analogous to Clairaut's; they have the form z =px + qy + f(p, q). () That the solution is z = ac + by + f(c, b) (2) can easily be verified. This is the complete integral, since it contains two arbitrary constants. It represents a doubly infinite system of planes. In order to obtain the general integral, put b = -(ca), where p denotes an arbitrary function; then z = ax + y~(a) ~+f ca, p(cac); differentiate this with respect to a,.0 = +-y '(ca) + '(c), and eliminate a between these equations. In order to obtain tle singular integral, differentiate z = ax + by+f(a, b) with respect to a and b, thereby getting the equations O=x+df 0=y+if; da db and eliminate a and b between these three equations. 'M 162 DIFFERENTIAL EQUATIONSCH. CH.II. Ex. 1. Solve z =px+ qy + pq. The complete integral is z = ax + by + ab. In order to find the singular integral, differentiate with respect to C and b; this gives 0 =x + b, =y + a; elimination of a and b by means of these equations gives z -xy. The general integral is the a eliminant of z = ax + yf(a) + af(a), 0 = x + yf' (a) -- +f (a) + f(a), where f denotes an arbitrary function. Ex. 2. Solve z = px + qy - 2/pq. 121. Standard III. To this standard belong equations that do not contain x or y; they have the form F((zp,q)=. (1) Put X for x + ay, where a is an arbitrary constant, alnd assume z =f(x + ay) =f(X) for a trial solution; then dz X dz dz aX dz 2=_ = = q= a —a dX x' dclX ' q dX y cldX Substitution in (1) gives F(z, dz dz \IdXi dX) which is an ordinary differential equation of the first order. The solution of (2) gives an expression of the form dz X = - (z, a), dX ' ' dz whence, (- = dX; ~integrating, f( (z a)= + b integrating, f(z, a) - X + b, ~ 121.] STANDARD II. 163 and hence, x + ay + b =f (z, a) is the complete integral. The general and the singular integrals are to be found as before. This method of solving equations of Standard III. can be formulated in the following rule: dz Substitute ap for q, and change p to c-,X being equal to cliX x + ay; then solve the resulting ordinary differential equation between z and X. Ex.1. Solve (1) z2(p2+q2+l)= 2. On putting ap for q, changingp to -X, and separating the variables, (1) becomes /Ya2 + 1 zz = dX. /c2 _ z2 Integrating, - x/a2 + 1 v/- z2 = X+ b; squaring, and substituting for X its value x + ay, (2) (a2 + 1)(c2 - 2) -(x + ay + b)2 This is the complete integral of (1), since it contains two independent arbitrary constants a and b. Differentiate (2) with respect to a and b, and eliminate a and b; there results z2 = 2, which satisfies (1), and is thus the singular solution. In order to find a general integral, substitute for b some function of a, and eliminate a from the equation. In particular, on putting b =- ak- h, (2) becomes (3) (a2 + 1)(c2 - z2)= {x - h + a(y - k)}2. (3) Differentiation with respect to a gives the equation 2 a(c2 - z2) = 2(y - k)x - h + a(y - k)}, which in virtue of (3) can be put in the form (4) {x - h + a(y - k)}{a(x- h)-(y- k)} = 0. (4) 164 DIFFE1RENT'IAL EQUATIONS. [Ci. XII. On eliminating a from (3) by means of the first component equation of (4), there appears the equation Z2 -= 2; and on eliminating a by means of the second component equation, there comes (a- h)2 +(y - k)2 + 2 c2. (5) The general integral is thus made up of the last two equations, which represent two parallel planes and a sphere. The planes andL sphere forml the envelope of the cylinders represented by (3). Equation (5) may also be regarded as a complete integral, if h and k be taken as arbitrary constants. (See Ex. 1, Art. 108 and Art. 112.) Ex. 2. Solve q2y2 = (z -px). This may be written ( ay Z -x- and putting dYfor y, dXfor A, (whence Y= log y and X= log x), y x the latter equation becomes which belongs to Standard III. Ex. 3. Solve 9(p2z + 2)= 4. Ex. 4. Solve p(1 + q2) = q(z - a). Ex. 5. Solve pz = 1 + q2. 122. Standard IV. To this standard belong equations that have the form fl(x, p) = f2(, q). (1) In some partial differential eqiiations in which the variable z does not appear, it happens that the terms containing p and x can be separated from those containing q andi y; the equation then has the form (1). Put each of these equal expressions equal to an arbitrary constant a, thus, fi(x, p),= a, f2(y ) =a; ~ 122.] STANDARD 1V. 165 and solve these equations for p and q, thus obtaining p = F1(x, a), q = F2(y, a). Integration of the last two equations gives z == f (x, a)dx + a quantity independent of x, and z = F2(y, a)dy + a quantity independent of y. These are included in, or are equivalent to z =fF (x, a) +fF2(y, a) + b, where b is an arbitrary constant. This is the complete integral, since it contains two arbitrary constants; the general integral and the singular integral, if existing, are to be founc as before. Ex. 1. Solve q-p +x-y= O. Separating q and y from 1 and x, q- y= -x. Write q-y =p-x= a; hence p = x + a and q = y + a; and therefore the complete integral is 2 z =( + a)2 + (y + a)2 + b. There is no singular integral; the general integral is given by the elimination of a between 2 =( + a)2+ (y a)2 + f (a) and 0 = 2(x + a)+ 2(y + a) +f'(a), f being an arbitrary function. Ex. 2. Solve p2 - q2 = -. z r i Hence zp2 - = zq2 - y. Put dZ for zdz. Ex. 3. Solve q = 2yp2. Ex. 4. Solve /p + /q-2 = 2 x. Ex. 5. Solve P2 + q2 = x+ y. Ex. 6. Solve z2(p2 + q2)= x2 + y2. 166 DIFFERENTIAL EQUATIONS. [CH. XII. 123. General method of solution.* It will be remembered that, in order to solve some of the ordinary differential equations of the first order in Arts. 24-29, another differential relation was deduced; and by means of the two differential relations, that were thus at command, the derivative was eliminated and a solution obtained. The general method of solving partial equations of the first order will be found to present some points of analogy to the method employed in the articles referred to. Take the partial differential equation F(x, y, z p, q)=. (1) Since z depends upon x and y, it follows that dz = dx +- q dy. (2) Now if another relation can be found between x, y, z, p, q, such as f(x, y, z, zp, q) =, (3) then p and q can be eliminated; for the values of p and q deduced froin (1) and (3) can be substituted in (2). The integral of the ordinary differential equation thus formed involving x, y, z, will satisfy the given equation (1); for the values of p and q that will be derived from it are the same as the values of p and q in (1). A method of finding the needed relation (3) must now be devised. Assume (3) for the unknown relation between x, y, z, p, q, which, in connection with (1), will determine values of p and q that will render (2) integrable. On differentiating (1) and (3) with respect to x and y, the following equations appear: * This method, comrionly known as Charpit's inethocl, in which the non-linear partial equation is connected with a system of linear ordinary equations, is due partly to Lagrange, but was perfected by Charpit. It was first fully set forth in a memoir presented by Charpit to the Paris Academy of Sciences, June 30, 1784. The author died young, and the memoir was never published, ~ 123.] GENERAL METHOD OF SOLUTION. 167 OF OF OF Fdp OF aOq_ ôx OZ ap ôx aq ôx OF OF OF ap OF aqF 0 ôy -z Op ay dq ay af f 0f ap af aq -o +y q+~+0 = 0, ôy az 'p 9y ôq ôy a- + dz q + $ Y + a- 0- -~ The elimination of -p between the first pair of these equations gives ax aO 0p +ay Oz a(O Pp dp Oz q aF aôp aq; and the elimination of -q between the second pair gives dy faOFOf aF Oaf (aF af OF Odf\ alO aF af a0 f 0) \ay aq aq ~ay Oz 9q aq z)+ y+ Op aq aq aJl On adding the first members of these two equations, the last bracketed terms cancel each other, since aq a2z Op. Ox aOxy -y' hence, adding and re-arranging, (aF O aF af O F 0 aN 0f O ( F qOF Of +- - - +q -+ r -qx a0z p ap ay Oz q a aq aOz +(- + (- F) f 0 (4) This is a linear equation of the first order, which the auxiliary function f of equation (3) must satisfy. This form has been considered in Art. 117, and its integrals are the integrals of 168 DIFFERENTIAL EQUATIONS. [CH. XII. (dp dq dz dx OF OF F F F OF OF F - OF +. +q - p __ q_ dx 0Z ay dz rP dq dp _ dy df (5) - F 0 ôq Any of the integrals of (5) satisfy (4); if such an integral involve p or q, it can be taken for the required second relation (3). Of course, the simpler the integral involving p or q, or both p ancl q that is derived from (5), the easier will be the subsequent labor in finding the solution of (1). This method is applicable to all partial differential equations of the first order; but it is often better to enquire whether the equation to be solved is reducible to one of the standard forms discussed in Arts. 119-122. The reduction and the subsequent integration by one of the special methods is generally, but not always, less laborious than the integration by the general inethod. By applying the general method to the linear equation and the standard forms, the integrals obtained in the preceding sections are easily obtained.? Ex. 1. Solve (1) p(q2 + 1)+(b-z)q =0. Here equations (5) Art. 123 reduce to C() dpd c dz __- x = dy pq q2 3pq2 +p +(b - z)q q2 + 1 - + b + 2pq The third fraction, by virtue of the given equation, reduces to dz 2pq2 From the first two fractions, there comes, on integration, q =ap, where a is an arbitrary constant. This and the original equation determine the values of p and q; namely, P /-(z-b)-1, g a(- b)-1. a * See Forsyth, Differential Equations, Arts. 203-207; Johnson, Differential Equations, Arts. 288-293. ~124.] EQUATIONS OF TIIE SECOND ORDER. 169 Substitution of these values in dz = dx + q dy gives dz = (x + dy) x/a(z - b)-1, where the variables are separable; this on integration gives 2 Va( - b)- 1 = x + ay + b. There is no singular solution; the general solution is obtained in the usual way. This equation comes under Standard III., and the ratios cliosen from (2) give the relation q = ap, which is used in the special method. Had there been chosen the equation formed by another pair of ratios from (2), say from dq _ dx q2 q2 + 1 another complete integral would have been obtained; namely, (z - ) { +- + a + + + + b =o. Ex. 2. Solve z pq by the general method. Ex. 3. Solve (p2 + q2)y= qz. Ex. 4. Solve the linear equation and the standard forins by the general method. PARTIAL DIFFERENTIAL EQUATIONS OF THE SECOND AND HIGHER ORDERS. 124. Partial equations of the second order. In this and the following articles,* a few of the simplest fornis of partial differential equations of the second order will be briefly considered; hardly more will be done, however, than to indicate the methods of obtaining their solutions. Some of these equations are of the highest importance in physical investigations. In what follows, z being the dependent variable, and x and y the independent, r, s, t will denote the second derivatives: * In connection with these articles read the introductory chapter of W. E. Byerly, Fourier's Series and Spherical Hlarmonics. 170 DlIFFERENTIAL EQUATIONS. [CH. XII. 2%z _ a 'z 0%2 r -~~, s= - -- dôx dx ay, ay2 There will be discussed linear equations only; that is, equations of the first degree in r, s, t, which are thus of the form Rr + Ss + Tt V, where R, 8, T, V are functions of x, y, z, p, q. The complete solutions of these equations will contain two arbitrary functions. In Art. 125 will be given sonie examples of equations that are readily integrable, the special method of solution necessary being easily seen; and in Art. 126 will be given a general method of solution. 125. Examples readily solvable. It is to be remembered that x and y, being independent, are constant with regard to each other in integration and differentiation. Ex. 1. Solve -2z - +a. ôx y y Writing it + a, dy y integration with regard to y gives p =x logy + ay + 0l(x), the constant with regard to y being possibly a function of x. Integrating the last equation with regard to x gives z = S fx log y + ay + li(x)}clx, = log y + a xy + 0(x) + /(y), the constant with regard to x being possibly a function of y. Ex. 2. Solve +- f (x) = F(y). x ady Ox Rewrite it, dp + pf(x) = F(y). Rewrite it, dy * See Art. 109, Ex. 5. ~ 125, 126.] Rr + Ss + Tt = V. 171 This equation is linear in p, and x is constant with regard to y; hence integration gives p = e-yf(x) i eYf(xF(y)dy + 0(x)1; and integration of tlis with regard to x gives z = { e-y)eyf(x)F(y)dy + O(x) }dx + V(y). Ex. 3. a = xy. Ex. 4. xr = (n - l)p. 126. General method of solving Rr + Ss + Tt V. On writing the total differentials of p and q, dp -= r dx + s dy, cq = sdx + t dy, the elimination of r and t by means of these from the given equation, Rr + Ss + Tt = V, (1) gives (R ctp dy + Tdq cx - Vdxdy) - s(R dy2 -S dxc y + Tdx2) = 0. If any relation between x, y, z, p, q will make each of the bracketed expressions vanish, this relation will satisfy (1). From R dy2 -- 8 dx dy + Tdx2 = 0 ] Rdp dy + Tdq dx - Vdx dy = 0 J and dc =c p dx + q dy, it may be possible to derive either one or two relations between x, y, z, p, q called intermediary integrals, and therefrom to deduce the general solution of (1). For an investigation of the conditions under which this equation admits an intermediary integral, and for the deduction of the way of finding the * These are called Monge's equations, after Gaspard Monge (1746-1818), the inventor of descriptive geometry, who tried to integrate equations of the form Rr + Ss + Tt = O, in 1784, and succeeded in some simple cases. The method of this article is also called by his name, 172 DIFFERENTIAL EQUATIONS. [CH. XII. general integral see Forsyth, Differential Equations, Arts. 228 -239. The statement of the method of solution derived from this investigation is contained in the following rule: Form first the equation R dy2 - Scx dy + Td2x = 0, (3) and resolve it, supposing the first member not a complete square, into the two equations dy - nzldx = O, dy - m2dx= O. (4) From the first of these, and from the equation R dlp cdy + Tdq dx - V7dx dy, (5) combined if necessary with dz = p dx + q dy, obtain two integrals uz = a, v, = b; then Ul = fl (vi), where f, is an arbitrary function, is an intermediary integral. From the second of the equations (4), in the sanie way, obtain another pair of integrals, u, = a, v2 = b; then t2 = -2 (y2) is another intermediary integral, f, being arbitrary To deduce the final integral, either of these internecdiary integrals may be integrated; and this must be done when mn = mn. 'When mn and m2 are unequal, the two intermediate integrals are solved for p and q, and their values substituted in dz = p dx + q dy, which, when integrated, gives the complete integral. Ex. 1. Solve r - a2t = O. (This equation is solved by another method in Art. 128.) Here the subsidiary equations (4) and (5) are (1) dy a dx = O, dy -a d(x = 0, (2) dp dy - a2dx dq = O. Hence y + ax - ci, y - ax = C2. ~ 127.] GENERAL LINEAR PARTIAL EQUATION. 173 Combining the first of equations (1) with (2), dp + adq = 0, whence p + aq = c'l = F1(y + ax); combining the second of (1) with (2), dp - adq = 0, whence p - aq = c'2 = -2(y - ax). From the last two integrals p [= -1(y + ax)+ F2(y - ax)], and q= [F(y + ax)- F2(y - ax)]. 2a Substitution of these values of p and q in dz = p dx + q dy, gives, on rearranging terms, dz = [Fl(y + ax)(dy + a dx)- F2(y - ax)(dy - adx)], 2 a which is exact. Integration gives z= 0(y + ax)+ p(y - ax). u a2 022= The equation in this example, - a2 - = 0, is a very important one ôt2 Jy2 in mathematical physics. It is called the equation of vibrating cords, sometimes D'Alemnbert's equation, from the name of the geometer who first integrated it in 1747.* It appears in considering the vibrations of a stretched elastic string, t being the time, y being ineasured along the string, and it being the small transversal displacement of any point. This equation also gives the law of small oscillations in a thin tube of air, for instance, in an organ-pipe. The functions 0 and & that appear in the general solution are to be determined from the given initial conditions. Ex. 2. ps - q = O. Ex. 3. xr + 2 xys + y2t = 0. 127. The general linear partial equation of an order higher than the first. A partial differential equation, which is linear with respect to the dependent variable and its derivatives, is of the form Ao + A - 0 + * + A a + Bo - + OxI OXn-l O y ayz On-1 +M +N + +Pz =f(, ), (1) Ox Oy * Jean-le-Rond D'Alembert (1717-1783), who first announced in 1743 the principle in dynamics that bears his name, was one of the pioneers in the study of differential equations. 174 DIFFERENTIAL EQUATIONS. [CH. XII. where the coefficients are constants or functions of x and y. On putting D for a and D' for -, this may be written ax ay (AoD + AD"-D' +.** + AD'" + * + MD + ND' + P)z =fy (, Y) (2) or briefly, F (D, D') z =f (x, y). (3) As in the case of linear equations between two variables (see Art. 49), the complete solution consists of two parts, the complemenntcay function and the particular integral, the complementary function being the solution of F(D, D') z = 0. (4) Also, if z = z, =, = z be solutions of (4), z = c1z1 + c2Z... + Cz.~ is also a solution. Other analogies between linear partial and linear ordinary equations, especially in methods of solving, will be observed ini the following articles. 128. The homogeneous equation with constant coefficients: the complementary function. All the derivatives appearing in this equation are of the same order, and it is of the form (AoD + D'D + AlD ' + AD't) z =f(x, y). (1) If it be assumed that z = (y + mx), differentiation will show that Dz = m4' (y + mx), D"z -= r,(n) (y + Mx), D'z = z (n) (y + mx), and, in general, that D"Ds'z = mr"+(rs) (y + mx). Therefore, the substitution of ~. (y + mx) for z in the first member of (1) gives (Aom" + Axn"-l +.. A,) ( (n) (y + mx). This is zero, and consequently, > (y + max) is a part of the complementary function if m is a root of ~ 128, 129.] THE HOMOGENEOUS EQUATION. 175 Aon^l + A m"-}.- *- + AI,-= 0, (2) which may be called thle caxiliary equation. Suppose that the n roots of (2) are m,, m2, *... m2, then the complementary function of (1) is z =- 1(y + m1x) + 2( + m( 2 +) A + ** (y + mfx), where the functions < are arbitrary. The factors of the coefficient of z in (1) corresponding to these roots are D -n mD', D - m2D', *., D- D,,)'; and these are easily shown to be commutative. (Compare Arts. 50, 54.) Since emD'p (y) (1 + mxD' + m2D2 D' + ) (y), M 22X2 = (y) +,mxq,(y) + 2 '/(yJ) +', = < (y + m2X), the part of the C.F. corresponding to a root m of (2) may be written e.X..D' (y). Ex. 1. z- a22z 0. (See Ex. 1, Art. 126.) aX2 dy2 Here (2) is m2 - a2 = 0, whence m has the values + a, - a. Hence the solution is z =- (y + ax) + (/y - ax). Ex. 2. Find tle C.F. of - + 3 -0+ 2 + y. ôx2 ôrx a 0 ay2 Ex. 3. Find the C.F. of 02- 2 - 6 0 = xy. dx2 ax y ay2 129. Solution when the auxiliary equation has repeated or imaginary roots. As in the case of equations between two variables (see Arts. 51, 52), further investigation is required when the roots of (2) Art. 128 are multiple or imaginary. The equation corresponding to two repeated roots m is (D - 1tD')(D - qLD') z = 0. On putting v for (D - mD') z, this becomes (D - mD') v = O, of 176 DIFFEIRENTIAL EQUATIONS. [CH. XII. which the solution is v = (y + mx). Hence (D - mD')z = ( (y + mx). The Lagrangean equations of this linear equation of the first order are dx = dy = z m < (y + mx) The integrals of these equations are y + mx = a, z-= x (a) + b; and hence, z = X(P (y + mx) + (y + mx). By proceeding in this way it can be shown that when a root m is repeated r times, the corresponding part of. the complementary function is r 1i(Zy+ rC) + ~X -22(y + mx) +.. +X<._i(y + mxn) + cP+ + ptix). When the roots of (2) Art. 128 are imaginary, the correspondiig part of the solution can be made to take a real form.* Ex. z 8-2-z 0 —,z 02z 3Z o Elx. 3 + O,- - = 0 ax3 Ox2 aY ax ôJ2 ~qy3 130. The particular integral. Equation (1) Art. 128 being expressed by F(D, D')z = < (x, y), the particular integral will be denoted b-y ( ' (' y), D D V being defined as that function which gives V when it is operated upon by F (D, D'). (Compare Art. 57.) By Art. 128, _ _ 1 1 F(D, DI) D - m,D' D - inD' D - ',,ID It is easily shown that it follows from the definition of 1 V( D) that these factors are commutative. The value of 1 D S(x, y) will now be indicated. For this purpose, D - -D' * See Johnson, Differential Equations, Art. 319; Merriman and Woodward, Higher Mathematics, Chap. VII., Art, 25. ~ 130.] TIE PARTICULAR TNTEGRAL. 177 it is first necessary to evaluate (D - mD') <' (x, y). From the latter part of Art. 128 it follows that e-xD'< (x, y) -= (x, y - mx); therefore, De-mD ' (x, y) = D< (x, y - mx). Direct differentiation shows that De-mxD'< (, y) = e-nD' (D - ïnD') (x, y). From equating the second members of the last two equations, and operating ipon these members with ce', it follows that (D - nmD')< (x, y) = eD'DpD (x, y - mx). That a similar formula D ' (x, y) =fD e - <D' (x, y - mx) (2) D - mD' D holds true for the inverse operator is easily verified. For, the application of D - LD' to both sides of (2) gives < (x, y) = (D - 'mD'2)e"D' (x, y - mx), - (D- mD')e D' (X, y), on putting (x, y) for 1p (x, y - mx); and, therefore, by Art. 128, < (x,?) = (D - mD')i (x, y + mx). But the second member of the last equation is also the result that would be obtained by putting y + mx for y in Dp (x, y) after the differentiation had been performed; and this would be b (x, y) froni the definition of given above. Hence D -D-< (x, y) can be evaluated by the following rule, which is the verbal expression of (2): form the function < (x, y - mx), integrate this wVith respect to x, and in the integral obtained, change y into y + mx. N 178 DIFFERENTIAL EQUATIONS. [CII. XII. The value of the second member of (1) is obtained by applying the operations indicated by the factors, in succession, beginning at the right. Methods shorter than this general method can be employee in certain cases, which are referred to and exemplified in Art. 132. Ex. 2 also shows such a case. Ex. 1. Find the particular integral of Ex. 2, Art. 128. The particular integral 1 1 1 (x+y) D2+3DD+2D'2 D D+2D' D+D' D+ 2 2D' D+2D' D+ 2 D' 1 92X3 ( 2 x2y) X3 Dx(y + 2x)= 2 ( x 2x)= D 3 2 2 3 Ex. 2. Evaluate 0(ax + by). In this case a short method F(D, DI) can be used in finding the integral. Since F(D, D') = DnF T-D anad D (ax + by)= and consequently D' D a F1D ) S( by) =, it follows that F(D (ax + by)= 0(ax + by)= (ax + by) F(D(, D Ia D l Dît(D F( ~p(l (b ) 0S.* (ax + by) (lx)n \a When b isaroot of F(D O, then = (D and the integral is b D + b)(yX)n; the latter Ip 0 D( - _, expression can be evaluated by the general rule. Ex. 3. Find the particular integral of a2u _ a2 -2u = x2. 0x2 dY2 Ex. 4. Find the particular integral of Ex. 3, Art. 128. ~ 131.] THE ANOX-IOMOGENEOUS EQUATION. 179 131. The non-homogeneous equation with constant coefficients: the complementary function. In order to find the complemlentary function of (3) Art. 127, that is, the solution of F(D, D')z = 0, (1) first assume z = cec7+k%. (This procedure is like that of Art. 50.) The substitution of this value of z in F(D,D')z gives cF(h, 7)ehx+ky. This is zero if F(h,k)=O; (2) and then z = ce7x+ky is a part of the complementary function. The solution of (2) for k will give valuesf(h), f2(h) *,,;fr(), if D' is of degree r in (1). The part of the solution of (1) corresponding to k =ff(h) is Ecle~+ft(7')Y, 8 indicating the infinite series obtained by giving c and h all possible arbitrary values; hence the general solution corresponding to all the values of k is z = Sclel7"x-il(7)Y + 'C2e7t+f2()Y +... + Cehx+fr(7l)Y This solution can be put in a simpler form when f(h) is linear in I7, that is, when k = ah + b. In particular this is true of the hoinogeneous equation, which is, of course, a special case of (1). Exs. 2, 3 illustrate these remarks. Equally well may (2) be solved for h in terms of k, and another form of the solution will be obtained, as in Exs. 1, 2. Ex. 1. 3z a-2z =0. ôa3 ay2 Here (2) is h3 - k2 = 0, whence k = hi, and thus the solution is z = cehx+hay, where c and h are arbitrary. Particular integrals are obtained by giving h particular values; for example, the values 1, 5, 4 for h give the particular solutions z = ex+y, z = e5s+52y, z ei-+i=y. If equation (2) be solved for h, the particular integral is ScekY+k'x. Ex. 2. 2az az az a2 6 O +3 a dx2 dx dy 9y2 Ox Jy Here (2) is 2 h2 - hk - k2 + 6 h + 3 k = 0, where the values of k are - 2 h, h + 3; hence z =- cieh(x-2y) + 2el(x+y)+3y - ZCleh(x-2y) + e3Yc.2e^(x+y). 180 DIFIFEBEN'IAL EQUAT1ONS. [Cil. XII. Since each of these series consists of terms having arbitrary coefficients and exponents, it can be represented by an arbitrary function. Consequently the solution can be represented by z - (x - 2 y) + e3Y(x + y). The equation above miight have been solved for h, the values being k k- 3. Hence, 2 z = clek(Y)+~eî+Y)-3x(y - + e-z/(x + y) is another way in which the solution may be written. Ex. 3. Solve Ex. 1, Art. 128 by this method. Here the values of h are ak, - ak, and hence Z = zcek(Y+ax ) += + Ca) + (y - ax). Ex. 4. Find the complementary function of a _ _ 3 +3 az= xy + e+2y. ax2 'y2 Ox xy Ex. 5. Find the complementary function of 2 3 + -3z = cos(x +2 )+ e. dx2 Ox Oy a Ex. 6. Find the complementary function of 2 - — z +- ~ + 3 - -2 z = ex-y - x2y. dX2 dy2 ôx ay 132. The particular integral. The particular integral can be obtained in certain cases by methods analogous to those shown in Arts. 60-64. It is easily shown, by the method adopted in Arts. 60-62, that er+y -= t (iax+hy; that F(D, 1)') F(a, b) sin (ax + by) = in (ax + by), F(D2, DD', D21) F(-a2, -ab, _-2) and similarly for the cosine; and that F(, ')x"ys can be F (D, D') evaluated by operating upon xys with [F(D, D')]-1 expanded in ascending powers of D and D'.* * For a full discussion, see Johnson, Diferential Equations, Arts. 328 -334. 132.] EXAMPLES. 181 Ex. 1. 2 Z -Z 2 z 2 +Z + 2 =e2x+3y + sin(2 + y) + xy. ôx2 ôx y Ôy2 ôx ay The complementary function, found by Art. 131, is (y - x)+ e-2x(2 x + y). The particular integral is h2-DD'-2h2 + 2hD + 2h' {e2x+sy 3 + sin(2 x + y) + xy}. D2 - DDI - 2 D12 q 2 D q- 2 Dl 1 e2ax+3y e2x+3y.ex3 D2 - DD' - 2 Dt2 + 2 D + 2 D' - 10 --------- 1 sin(2 x + y) D2- _DD- 2 2'2 + 2 D) 2 Dl _- 1 - sin(2x+ y)-lD- D' 2(ID-) Dsin(2x -y)2 D2 D2sin(2x + y) 2(h+D') 2h2-h'/2 _ cos(2 x + y) 6 1 1 1 xy — ~ xy 2 - DDI - 2 2 + 2 + 2' + D' D Dl -2D +2 =1 -- 1 D-2D' +(D —2D')2 2 D+D' 2 4 y I 1 (~:y - l y + x - l) 2 D -+Dl' -20~~ ~~(6-6Xy —y —29x —1 ) =1 *l (1_ D )(xY- Y+x-1)=D(xy/ 'y 3 -x (6xy - 6y - 2x2 + 9 x - 12). 24 Therefore, the general solution is = ( - x) + e-2x(2 x + y) - -Ie2+3y -cos(2 x + y) 10 6 + -(6xy - 6 y - 22 + 9x - 12). 24 Ex. 2. Solve Exs. 1, 3, 4, Art. 130, by the shorter methods. Ex. 3. Find the particular integrals of Exs. 4, 5, 6, Art. 131. 182 DIFFERENTIAL EQUATIONS. [CH. XII. 133. Transformation of equations. The linear partial differential equation with variable coefficients, like the linear equation between two variables, inay sometimes be transformable into one having constant coefficients. In particular, an equation in which the coefficient of any derivative is of a degree in the independent variables equal to theé number indicating the order of the derivative, is thus reducible. This is illustrated by Ex. 1. (Compare Arts. 65, 71.) Ex. 1. x2 y2-y x = 2. ôx2 dy2 y Oax On assuming u = log x, v = log y, the equation takes the form 02z 2 0= 0, Ozt2 av2 of which the solution is z = (u + v)+ V(u - v). The substitution of the values of uz, v, gives z = 0(log(xy)) + log ) =f(xy) + F ( ) Ex. 2. x2 2z- yx4 2z +4y2 Z + 6 y a 3y4. 0x2 axay Oy2 Oy Ex. 3. -1z -2 x2x2 x3 x d y2 Y2 y3 y 134.* Laplace's equation: V2v = O. The equation ô2v 02 v 02v + + = 0, (1) usually written V2 O = 0, and commonly known as Laplace's t equation, is one of the equations most frequently met in investigations in applied mathematics, appearing, as it does, in discussions on mechanics, sound, electricity, heat, etc., especially where the theory of potential is involved. * Arts. 134, 135, 136, are merely notes. t Because it was first given, in 1782, by Pierre Siméon Laplace (1749 -1827), one of the greatest of French mathematicians. ~ 133, 134.] LAPLACE'S EQUATION. 183 For instance, if V be the Newtonian potential due to an attracting mass, at any point P(x, y, z) not forming a part of the mass itself, V satisfies (1); again, if V be the electric potential at any point (x, y, z) where the electrical density is zero, V satisfies (1); t and, to give one more instance, if a body be in a state of equilibrium as to temperature, v being the temperature at any point, d- = 0, and v satisfies (1). If f(x, y, z) denote any value of v that satisfies (1), f (x, y, z) -= in the first two instances is called an equipotential surface, and in the third an isothermal surface. On changing to spherical co-ordinates by the transformation x = r sin 0 cos s, y = r sin 0 cos s, z = r cos 0, (1) becomes: 02v 1 02v 2 av cot 0 av 1 O2 + — + r = O, (2) ar2.- 02. Or.'2 oa r Sin2 B 2 which may be written r2 a r2Ôr+/ siO O, 0 sinv O a02 =O; (3) r (-i- sin 0 0 dB) silln 0ô(Ps and if tu = cos 0, it will take the form +2(V ( 2)}v+ 1 02v =. (4) ar2 9 p I. - L a(P The subject of Spherical HaIcrmonics is in part concerned with * B. O. Peirce, Newtonian Potenztial Funzction, Art. 28; Thomson and Tait, Natuiral Philosophy, Art. 491. t W. T. A. Eintage, Mathematical Theory of Electricity and Maegnetism, p. 14. ~ Todhunter, Digferential Calclhlus, Art. 207; Williamson, Differential Calculus, Art. 323; Edwards, Differential Calculus, Art. 532. The equation as given by Laplace was in the form (2). 184 DIFFERENTIAL EQUATIONS. [CI. XII. the development of fiinctions that will satisfy this equation.: A homogeneous rational integral algebraic function of x, y, z of the nth degree, that is, a function of the form rnf(0,,) in spherical co-ordinates, which is a value of v satisfying (1), is called a solid spherical harmonic of the nth degree; and f(0,,) is called a surface spherical h'armonic of the nth degree. Spherical harmonics are also known as Laplace's coefficients.If v be independent of ~, (3) reduces to 2+ - a sin 0 - =O. (5) or2 sinO 00 -0 j/ On putting v - '"P, where P is a function of 0 only, and changing the independent variable 0 by means of the relation J = cos 0, (5) becomes (1 _- 2) t +n(n + 1)P =, (6) which is Legendre's equation, Art. 83. A function that satisfies (6) or (5) is called a surface zonal haricmonic. A particular class of zonal harmonies is also known as Legendrean coefficients.1 For a treatment of spherical harmonies, see Byerly, Fourier's Series and Spherical IHacr' onics, Chap. VI., pp. 195 -218; and of zonal harmonies, see the same work, Chap. V., pp. 144-194. In special cases (1) and its solution assume simple forms; two of these will now be shown. See Williamson, Dif'erential Calculus, Chap. XXIII., Arts. 332 -337; Edwards, Differential Calculus, Art. 189; Laimb, BHdroclynamics, Ed. 1895, Arts. 82-85; Byerly, Foutrier's Series ies ad Spherical Harmonies. t So called after Laplace, who employed them in determining V in a paper bearing the date 1782. t After Legendre, who first introduced them in a paper published in 1785. Legendre's work in this subject, however, was doie before that of Laplace (Byerly, Fourier's Series and Spjherical Harmonics, Chap. IX., p. 267). See Ex. 5, Art. 82. ~ 135.] SPECIAL CASES. 185 135. Special-cases. In the first instance given in Art. 134, suppose that the attracting mass is a sphere composed of concentric shells, each of uniform density. Here v obviously depends only upon the distance of the point P from the centre of the sphere, and hence (2) Art. 134 reduces to dv + 2 di =o, (1) dr2 r dr which on integration gives v=A+ -B. (2) Equation (1), in which v depends upon r alone, can be obtained directly front (1) Art. 134 by means of the relation = + y2 + z2. For, ôv_ dv dOr x dv Ox dr x r dr' ô2v 1 dv x 2 2 d2v ax2 r dr '3 dr r ' dr2 and on finding similar values for a, and adding, there results (1). y For the discussion and integration of (1) from the point of view of mechanics, see Thomson and Tait, Natural Philosophy, Vol. I., Part II., p. 35. If the point P in the second instance of Art. 134 be outside of a uniformly electrified sphere and at a distance r front the centre, obviously av = 0 and v _ 0; and equations (1) and (2) follow as before. For the interpretation and application of this result, from the point of view of electricity, see Emtage, Mathematical Theory of Electricity acd ifagnetism, pp. 14, 35, 37. Again, suppose that the attracting body in the first instance in Art. 134 is made up of infinitely long concentric cylindrical shells, each of uniform density, the z-axis being the common axis of the cylinder; or, that in the second instance P is a point 186 DIFFERENTIAL EQUATIONS. [CH. XII. outside an infinitely long conducting cylinder nniformly charged with electricity, the z-axis being the axis of the cylinder. Since in these cases v clepends only upon the distance from the axis of the cylinder, that is, upon x2 + y2, (1) Art. 134 reduces to dav (IaV il-t+ = O, dr2 dr which on integration gives v = Alog -; or = C - A log r, For discussion of these and other special cases, see the works referred to in the former part of this article, and also B. O. Peirce, Nezvtoniact Potential Function. 136. Poisson's equation: 'v =-4 7rp. If in (1) Art. 134 the second member be - 4 7p, p being a function of x, y, z, then there appears the equation - + 2 + 2-4r 4 rp, (1) ôx2 y'2 ôz2 which is known as Poisson's equation.* An example of its occurrence is the following: t If p be the density of matter at the point (x, y, z) in the first instance in Art. 134, equation (1) Art. 134 takes the above form. In the case of the sphere described in Art. 135, the equation becomes d2e 2 dv + =- 4.p, dr,2 r dr and the first integral is r 42 dv = — 4 pCdr - M dr ~ where M denotes the whole amount of matter within the spherical surface of radius r. In the case of the concentric cylinders, the equation becomes * So called from Siméon Denis Poisson (1781-1840), who thus extended Laplace's equation. t See Thomson and Tait, Natural Philosophy, Art. 491. ~ 136.] EXAMPLES. 187 d2v 1 dv - = — 4 tp, dr2 r dr and the first integral is dv r - = c - 4 prdr. dr EXAMPLES ON CHAPTER Xll. 1. xp + yq = nz. 4. (a - x)p (b - y)q = c - z. 2. (y2+z2-x2)p-2xyq+2xz=O. 5. (y+ z)p +( z+ x)q =x+y. 3. x -+y + t -=a +. 6. a(p + q) - z. Ox Oy 9t t 7. (yx - 2 4)p + (2 y4- x3y)q = z(3 - y3). 8. z-xp - yq = a /x2 y2+ - 2. 14. x2p=- yq2. 9. (x2-yz)p + (Y2- X)q =2 —Xy. 15. p2 q2 = npq. 10. (y - z)p + (z - )q x- 16. z - px - qy cl p2 + q2. yz zx xy 11. cos(x + )p + sin(x + y)q = z. 17. "/ + V = 1. 12. p2 =z(1- pq) 18. q=xp+p2 13. q =(z + px)2. 19. p(l + q) =gz. 20. Find three complete integrals of pq =px + qy. 21. (x2 + y2) (p2 + q2) = 1. 22. pq = xmynzl. 23. (x + y)(p + q)2 + (x- y)(p - q)2= 1. 24. (y - x) (qy - px) = (p -q)2. 25. (p + q) (px + qy) = 1. 26. xr +p =9. 27. s2+- -. 28. q2r-2 pqs + p2t =0. 29. q(1 +q)r -(p+ q + 2 pq)s + p(l +) )t=0. 30. yr = (n - 1)yp + a. 33. p + r xy. 3 21 d z 34. 31. - + a f()= F(y) 34. s = y. 0x2 ax 32. xr - p =xy. 35. r - (a + b)s + abt = xy. 36. (b + cq)2r - 2(b + cq)(a + cp)s + (a + cp)2t = 0 37. s+ =. 38. 37. s + ~~ p - ay3. 38. rq- s=15xy2. 1 - y2 9 MISCELLANEOUS NOTES. NOTE A. A system of ordinary differential equations, of which a part or all is of an order higher than the first, can be reduced to a system of equations of the first order. Take the single differential equation of order i d"y 2 d"-y dy ) f Y...., o, (1) Jdx-' dx-l' (lx' and put yi, = Y Y d /2, -., = Y,-. Then (1) can be replaced by dx dx2 dx1 -- the following system of n equations of the first order, dxy dx dy2 =Y2, dx / 1 Yn-1, Yi -2, l, Y, x)= 0. Again, suppose that there are two simultaneous equations, f (XY dY d2y d3y z dX d2z = o, dxd' dx2' dX3' dx' dx2] f(X, Y, Y d2y d3y dx Z2]0. dx dx2 dx3 dx dx'2 On putting dy- =yl y, dY - z= Y =1, these two equations can be dx dx2 ldx 1890 190 DIFFERENTIAL EQUATIONS. replaced by the following equivalent system of equations of the first order: dy _Y dx ldy dy- Y2, dx dz — = Zl, dx fA X, y, Y1, Y2, IY, Z, Z, z~-~ dz] dy2, dz1, f(x, y, yi, y2/, z' ' - d = 0 dx dx It is evident that any system of ordinary differential equations can be reduced in this manner to another equivalent system, where there will appear only derivatives of the first order. NOTE B. [This Note is supplementary to Art. 1.] The Existence Theorem. Following is a proof of the existence of an integral of an equation of the first order.* Suppose that the differential equation p(y', y, x)- 0, where y' stands for ly, is put in the form y =-f(x, y), (1) which is always possible. This proof is limited to the case where f(x, y) is a function which can be represented by a power-series t ao + alX + a2y -t- a32 + a4Xy + Ca5y2 + * + Xmyxy " +..., in which the a's are all known, since f(x, y) is known, and which converges for I xlr, Iy t, say. (The symbol Ixl denotes the numerical value of x.) * This proof is taken from notes of a course on differential equations given by Professor David Hilbert at Gottingen. t This is by far the most important case, since in the higher mathematics such functions are almost exclusively dealt with, and in applied mathematics they are universally used for approximations. MISCELLANEOUS NOTES. 191 It is to be shown that there is a convergent series y = ao +aix+ a2x2 +. (2) which identically satisfies y =f (x, y)= ao + aix + a2y + a3x2 + a4Xy + asy2 + * — + aix4my +...; (3) and which also satisfies a given initial condition, say, that y = Yo when X = X-.* That y = 0 when x = 0 may be taken for the initial condition without any loss of generality. For, on substituting x1 + xo for x and Yi + yo for y iln (1), it becomes Yi' = 0(xi, Yi); (4) and it is evident that for yi = ao' + al'x + a2lx2 + * to identically satisfy (4) and the initial condition that yl = 0 when x1 = 0, is the same thing as for (2) to satisfy (3) and the initial condition that y = Yo when x - xo. Hence the initial condition may be taken iin this form at the beginning; and for this it is both necessary and sufficient that ao in (2) be zero. It will now be shown (a) that there is one and only one series, y = aix + a22 +.., (5) which satisfies (3) identically; and (b) that within certain limits for x this series is convergent. On transforming the series in (3), which has been supposed convergent for Ixl<r, lyl\t, by putting x = rxl, y = tyl, equation (3) takes the form Y' =f(ixl, tyi)= ao' + al'xi + aG'yl + a31xl2 + a ~4'ly + * --- The second member of this equation is a convergent series, and converges when xi = y, =1; and, therefore, ao' + ati + a2C + *.. converges. This shows that the absolute value of each a' is not larger than a certain finite quantity A, say. The substitution just made for x and y does not make any essential change in the problein, and hence it might have been assumed at first that the a's of (3) were each not greater than A. In what follows the a's are accordingly regarded as not greater than A. * If an initial condition be not made, then an infinite number of series can be found which will satisfy (3). 192 DIFFERENTIAL EQUATIONS. If (5) satisfies (3), the value of y and y' derived from (5), when substituted in (3), must make the latter an identity; and, therefore, ai + 2 a2x + 3 a3x2 + *.. =a + a + x + a2(ax + a2x2 + **) + a3x2 + a4x(alx + a2x2 +. ) + a5(a1x + a2x2 +...)2+.. is an identical equation. Hence ai = ao; 2 a2 = ai + a2al, whence a2 = al + 2ao 3 a3 = a3 + 02a2 + a4al, 2 whence as = ~3 + ( (al + a2ao) + a4ao; and similarly for a4, a5, *... It is evident that all the a's can be determined as rational integral functions of tie a's; and it is also to be noticed that all the numerical coefficients in the expressions for the a's are positive; and, therefore, the a's will not be diminished if each of the a's is replaced by A. From the method of derivation it is evident that (5) with the a's determined as above identically satisfies (3). It has still to be determined whether this series is convergent. On replacing each of the a's in (3) by A, a quantity not less than any one of the a's, there results y = A(1+ x+y+- 2 + xy- y2+ 3 +- x2y...). (6) The integral of this equation is found by replacing each of the a's that occur in the expressions for the a's of (5) by A. None of these latter coefficients are diminished by changing each of the ~'s to A, as pointed out above; hence, if the integral of (6) is convergent, the integral of (3) is also. Now solve (6) directly. On factoring the second member, the equation becomes y'=A(1 + x +...)(1 4+ y 2+ * ), 1 1 1-x 1-y (lx Therefore, (1 - y)dy = A 1-x whence, on integration, y - y2 - A log(l - x) + cl. Therefore, y 1 + [2 A log( - x)+ c + 1]~. Here c must be determined, so that the initial condition be satisfied, namely, that y = 0 when x = 0; therefore 0 -1 c+1. Hence the square root must have the minus sign, and c must be zero. MISCELLANEOUS NOTES. 193 Therefore, y= l-[1+2Alog(l-x)]=l-[1-2 x+ +...)] (7) x2 x3 The series x + + + * converges for x < 1; hence the square root 2 3 ( 2 X3 \ of 1 -2 A x+ +2 3+ + -) converges for Ixl< 1; and hence the value of y in (7) is finite; and, therefore, the value of y in (5) is finite for x within certain limits. Note A showed that an equation of order n can be replaced by a system of n simultaneous equations of the first order, each containing an unknown function to be found. In the case of the equation of order n, the proof of the existence of integrals is made for this equivalent system instead of for the single equation of the nth order; the proof can be carried through in much the same way.-' The method of proof given above is known as "the Power-Series method." Historical Note.t - Augustin Louis Cauchy (1789-1857) of Paris, who was one of the leaders in insisting on rigorous demonstrations in mathematical analysis, gave the two first proofs of the existence theorem for ordinary differential equations. The first proof was given for real variables in 1823 in his lectures at the Polytechnic School in Paris; the second was given in 1835 for complex variables in a lithographed memoir. He was also the first who proved the existence of integrals of a partial differential equation. The first of the two proofs was published in Moigno's Calculus in 1844; this may be called " the method of difference equations"; it las been developed and simplified by Gilbert in France and Lipschitz in Germany. In his second method Cauchy employed what he called "the Calculus of limits." This method has been developed by Briot and Bouquet, and Méray in France, and Weierstrass (1815-1897) in Germany. (The proof given above follows Weierstrass' exposition of Cauchy's second proof.) A new proof, that by "the method of successive approximations," was given by mile Picard of Paris in 1890.t -Leo Koenigsberger, Theorie der Differentialgleichungen (Leipzig, 1889), p. 27. t For many historical notes and references relating to the existence theorem see Mansion, Theorie der Partiellen Dlfferentialgleichungen, pp. 26-29. t For an English translation of this proof made by Professor T. S. Fiske, see Bulletin N. Y. Math. Soc., Vol. I. (1891-1892), pp. 12-16. o 194 DIFFERENTIAL EQ UATIONS. In the Traité d'Analyse of É. Picard, t. II., pp. 291-318, will be found, besides the author's own proof just mentioned, Cauchy's first and second proofs, the latter as modified by Briot and Bouquet; and Madame Kowalevsky's proof * of the existence of integrals of a system of partial differential equations. (A knowledge of the theory of functions of a complex variable is necessary for the reading of some of these proofs.) t NOTE C. [This Note is supplementary to Art. 3.] The complete solution of a differential equation of the nth order contains n arbitrary independent constants. Let y', yft,... denote the first, second, *.. derivatives of y with respect to x, and y(0), y'(O), y"(0),... denote tle values of y, y', yl,... when x O. First, let an equation of the first order be considered; and suppose that the solution of F(y', y, x) = 0, (1) when expanded in ascending powers of x is y =- c + cx + 22 +.... (2) Note B shows that the solution can be thus expressed. But y(x) = y(O) + y'(0) x + 1 yt(O)x2 +... (Maclaurin's Theorem); (3) and therefore c = y(0), ci = yt(0), c2 = yl/(0),.... Now c =y(0) cannot be expressed in terms of anything known or determinable. Ilowever, cl = y'(0) can be determined, for F(yt, y, x) = O holds true for all values of x, and hence for = 0; therefore F{y'(0), y(O), 0} = 0, that is F(cl, c, 0) = 0. This determines c1 in terms of c. Equation (1) may be solved for y', thus, y' =f (y, x); (4) then, on differentiation, * Crelle, Vol. 80. (Memoir dated 1874.) Madame Sophie de Kowalevsky (1853-1891) was professor of higher mathematics at Stockholm (1884-1891), and received the Bordin prize of the French Academy in 1888. t For this Note, I am indebted to notes of lectures by Professor Hilbert at Gôttingen. MISCELLANEOUS NOTES. 195 y" = afy + f'; therefore y(O) y ()+= [l = Oy ax ôx Ox=o This determines c2 in terms of c and ci, and from this c2 can be found in terms of c alone. Another differentiation and the substitution of x = Oin the result will give an equation by means of which y"t'(O), and thus C3 also, can be expressed in terms of c; and similarly for the constants C4, C5, -... Therefore all the constants except c are determined; that is, the differential equation of the first order has one arbitrary constant in its general solution. In the next place, let an equation of the second order be considered. Put the equation p(y", y', y, x)= 0 into the form y" = f(y', y, x); (5) and suppose that the solution is y = c + clx + c2x2 +.... Determination of the values of c, cl, c2, *.., as before, gives c = y(O), ci = y'(O), 2 = y"(O),.... But, from the given equation, y"(0) = f{y'(O), y(0), O}; and this determines c2 in terms of c and Ci. On differentiating (5) and putting x = 0, there is obtained y(' L'= yY" + f yL + Ay o=__, F(cc, C2) F{,ci, Cf1o, c,,0)}; and hence c3 is found in terms of c and cl. By proceeding in this way, the values of all the other coefficients can be obtained in terms of c and Ci; but it will not be possible to obtain any information about c and ce. The solution of (5) will therefore contain two arbitrary constants. The proof of the theorem for equations of higher orders is made in exactly the same way as has just been used in the case of equations of the first and second orders. NOTE D. [This Note is supplementary to Art. 4.] Criterion for the Independence of Constants of Integration. In Art. 4 an example has been given of an integral in which there are apparently two constants of integration, but in reality these two are equivalent to only one. The question thus arises, how is it to be determined whether the constants of integration are really independent? 196 DIFFERENTIAL EQUATIONS. In the case of a solution of an equation of the second orcer having the form y = -(x, cl, c2), the criterion that the constants cl, c2 be independent, by which it is meant that this solution be not reducible to the form y = {x, f(cl, c2)}, in which there is really only one arbitrary constant f(cl, c2), is that the determlinant 020 * 02~ acl dx Oc2 dx be not equal to zero. For, suppose that 0(x, cl, c2) can be put in the forin {x, f(cl, c2)}. On forming and expanding the above determinant, there results aw af O2 a0f O2P df d. af af Oci x df ac2 Ox Of Oci af c2 which is identically zero. * Conversely, if the determinant be identically zero, then 0(x,, c, 2) must be of the form p{x, f(cl, c2)}; that is, 0(x, cl, c2) will not vary, no matter how cl and c2 are varied, so long as f(ci, c2) is assigned sonie particular constant value. On writing p, q, for a-, -, the condition that the determinant be aci aC2 zero takes the form p - q -=0, whence on integration P - a conO x Ox q stant; that is, P is independent of x, and hence can only involve cl and c2. Take where L, M, are fnctios f, and c2. Take where L, M, are functions of c1, c2. ilence M0 - L * Now differentiation of 0(x, cl, c2) gives c1 1 C2 d0 dx + 0 cdcl + I dc2 = 0dx + (Ldci + Mlci2) -1- dx acil c2 dx L ocl But Ldci + Mdc2 has an integrating factor,u, such that 1uLdcl +,u-tIdlc2 is a complete differential of the form dfc(ci, c2). Therefore do = c dx + 1 c- clf(ci, c2); hence (x, c1, c2) will Ilot dx -,L Odc vary, no matter how ci, c2 are varied, provided only that they satisfy the condition f(ci, C2)= a constant. Hence the necessary and sufficient condition that cl, c2 be really independent in 0(x, c1, c2) is that the above determinant be not equal to zero. * This part of the proof is due to Professor McMahon of Cornell University. MISCELLAINEOUS NOTES. 197 More generally, the criterion that the n parameters cl, c2, *., cn in f(x, c1, c2, *., Cn) be independent is that the determinant af af af dCi dc2 dc, a2f a2f 2f dcl dx dc2 dx dc, ax anf anf dci dx-l1 cCndxn-1 be not equal to zero. NOTE E. [This Note is supplementary to Art. 12.] * Proof that Pdx + Qdy is an exact differential when aP_ ady dx Let jPdx-=V, then adV d2; v= _ ax dx dY OY dx therefore dQ d (dV dx dx \ ay Hence Q -= + 0' (y), where o'(y) is some function of y. Therefore ay Pdx + Qd y = + V + x da jy + (y)y = d [ V+ p(y) ], an exact differential. NOTE F. [This Note is supplementary to Art. 49.] On the criterion that n integrals y1, 22, y*-, Yn of the linear differential equation dyz + l dlz,-' + + Py = (1) be linearly independent. * I am indebted for this proof to Professor McMahon, of Cornell. 198 DIFFERENTIAL EQUATIONS. Before proceeding to establish the criterion, it may be remarked that if there be a linear relation alYl ~+a2Y2 +* = O, (2) where al, a2,... are constants, existing between all or any of the integrals Yl, Y2,,, Yl, then the integral y = clyl + c2y2 + + cny, in virtue of (2), may be written ( 2\ ( 3 a y-= c2 - cl- )Y2 + C3 - Cl-y3 + *. + c - C 1l- y. \ aij ai 7 al/ This expression does not really contain more than n - 1 arbitrary constants, and therefore is not the general integral. Form the determinant Y Y2... Yn yi' y21... yn yl(n-1) y2(n-1)... y,,(n-1) where the elements of each row below the first are the derivatives of the corresponding elements in the row above them. This determinant is known as the functional letermilnant of Yl, Y2, **, Yn, and will be denoted by R. If there be a relation such as (2) between the integrals yi, Y2, **, Yn, then the elements of one of the columns of R are formed from several other columnns by adding the same multiples of the corresponding elements of these other columns; and, consequently, R will be identically equal to zero. Conversely, if B = 0, there will be a linear relation of the form (2) between the integrals y, y,, * —, y,,. Since R = O, the determinant must be reducible to a form wherein all the elements of one column are zero; that is, there must be certain multipliers X1, X2, *.., X,, such that NiYi + 22Y2 + + - \ Nn i 0 Xlyl + X2y2' + + XJyn, = 0 ]lYlt + X2Y21 + '" + Xl,,ry = 0.... XlYl(n-l) + X2,y(n-i) +...+ 4lyn(n-l)= O 0 Differentiation of each of these equations and subtraction of the one next following gives MISCELLANEOUS NOTES. 199 XliYl + X2t'2 + ~ + XInt = O Xl'yl' + X2'y2' +.*. + XyL = O i... (4) Xlfyl(n-2) + X2/y2(n-2) +... + Xdny(n-2) = 0 Xlyl(n-1) + X21y2(n-1) + + Xnyn(n-l) - + Xly(") 4 X2y2(n) + + X** yn(">) But the second line of the last equation is zero. This may be seen by substituting yl, Y2, **, Yn in (1), multiplying the equations thus formed by X1, X2, *-, X,, respectively, adding their first members, and taking account of (3) in the result. From (3) and (4) it follows that XII _V.21 _' k'... _ Xi k2 Xn Suppose that each of these fractions is equal to p, say. It follows from integration that Xi = aieSPd, X2 = a2eSpcl,.X, aneJSPd, ac, a2, ', c, being the constants of integration. On substituting these values in the first of equations (3) and dividing by the common factor ejPdx, there appears the relation ayli + a2y2 + * ' + anyl, = 0, which is thus a consequence of R being equal to zero. Therefore, the necessary and sufficient condition that Yl, Y2, **, Y, form a system of linearly independent integrals, or a fundamental system of integrals, as it is sometimes called, is that the determinant R do not vanish identically. NOTE G. The relations between the coefficients of a linear differential equation and its integrals. In close analogy to the problems of forming the algebraic equation of the nth degree when its n roots are given, and of finding the relations existing between its coefficients and the roots, are the problems of forming the linear differential equation of the nth order, of which n independent integrals, in other words the general solution, are given, and of finding the relations between its coefficients and the integrals. Let?y, Y2, ', Yn be n linearly independent functions of x. It is required to form the differential equation which has these functions for its integrals; in other words, to form the equation which has y = clyl + C2y2 + *. + Cnyn (1) 200 DIFFERENTIAL EQ UATIONS. for its general solution. The differential equation is formed by eliiiinating cl, c2, *.., c,. from the given integral by the method shown in Art. 3. By differentiating n times there is obtained the set of n + 1 equations, y, cll + C2y 2 + c *2 + c,,yn y, = Clyll + Clylt + *+. + cyn' yl(n) = Ciyl(n) + C22(.l)+ - + Cnyn(n) From this the eliminant of the c's is found to be Y Yi YV2 ' Yn Y' Yi' yt... Yn, 2) 11 MJ1/ 2t * t 82t' = ~, (2) y(n) yl(n) y2(n)... y,(n) the differential equation required. Now suppose that the differential equation having the integrals yl, y2, **., yn is in the former diy + p n-1y + P y +. + P () O ( ) d~y * r1 i + P2 O~. ~ + P__(Y) - 0. (3) dxn dxn-l dx~'l-2 On denoting the minors of y, y',..., y(n) in (2) by Y, Yi,..,, Yn, respectively, (2) on expansion becomes y d, y -ly dy d î_- d- + o.( +(- 1)ny = 0.4) dx~. dîXn-1 Comparison of (3) and (4) shows that P- Y= y, P2: =..., Pn =<(- l)~. Y.n Yn Yn It will be observed that Y, is the determinant E of Note F. *In particular, since differentiation will show that Yn-_ dx 1 P1i - dx; (^a; Y, d x and hence Yn = e-'Pdx. NOTE H. [This Note is supplementary to Art. 102.] On the criterion of integrability of Pdx + Qdy + Rdz = 0. It has been shown in Art. 102 that the necessary condition for the existence of an integral of Pdx + Qdy + Rdz = (1) * This deduction is due to Joseph Liouville (1809-1882), professor at the Collége de France. MISCELLANEOUS NOTES. 201 is that the coefficients P, Q, R, satisfy the relation rQU aR)~Q(pt, )~B +,J(i 2) =0. (2) p ôay "_x z ) + a(y _z) = It will now be proved that this condition is also sufficient, by showing that an integral of (1) can be found when relation (2) holds. Substitution shows that, if relation (2) holds for the coefficients of (1), a similar relation holds for the coefficients of 1zPdx + /Qdy + uRdz = 0, (3) where u is any function of x, y, z. If Pdx + Qdy is not an exact differential with respect to x and y, an integrating factor u can be found for it, and (3) can then be taken as the equation to be considered. Hence there is no loss of generality in regarding Pdx + Qdy as an exact differential. Let (Pdx + Qdy) = V(x, y, z), then the solution of Pdx + Qdy = O is V(x, y, z) = c. (4) Hence Vdx + zV dy + Vdz = 0, 3x rz )z that is, Pdx + Qdy + -Vdz = 0. (5) (In the first equation of (4) x and y alone are supposed to vary; in (5) x, y, z, can all vary.) Since (5) has an integral V(x, y, z) = c, a relation of the form (2) must exist between its coefficients; therefore p( o-' + Qf 2V _= V P _Q =o. (6) \ôz azdy \dôzadx a / \ ay ax The subtraction of (6) from the given relation (2) gives, the last bracket in each being zero since Pdx + Qdy is an exact differential with respect to x and y, p( a2v aRn a2o v a_ R \z ay dy/ \ zdx dx whence, P - d V - Q ( -R)=O. ayd / dx- / Since Pdx + Qdy = O when x and y alone vary, the latter may be written -( —1)\ dy++ ( —(V x=. ôy\ 6 x az 202 DIFFERENTIAL EQUATIONS. This has to be satisfied for all values of x and y, when x and y alone vary; hence, V -_ is independent of both x and y, and, therefore, is a dz function of z alone. Now (1) may be put into the form Pdx + Qdy + a- dz- -R dz, each side of which is an exact differential; and integration gives V(x, Y, z)= f a Hence (2)* is both the necessary and sufficient condition that (1) have an integral. t NOTE I. Modern Theories of Differential Equations. Invariants of Differential Equations. The two modern theories of differential equations are: (a) The theory based upon the theory of functions of a complex variable; (b) The theory based upon Lie's theory of transformation groups. The study of differential equations, until about forty years ago, was restricted to the derivation of rules and methods for obtaining solutions of the equation and expressing these solutions in terms of known functions. Even at the beginning of the present century, however, * Of course this criterion is included in the criterion for the general case of p variables, the deduction and proof of which is to be found in Forsyth, Theory of Differential Equations, Part I., pp. 4-12. (See footnote, p. 138.) See Serret, Calcul Intégral (edition 1886), Arts. 785-786. t Two historical articles that the student would do well to consult are: T. Craig, " Sorne of the developments in the theory of ordinary differential equations between 1878 and 1893," Bulletin of N. Y. Math. Soc., Vol. II. (1892-1893), pp. 119-134; D. E. Smith, " History of Moder Mathematics" (Merriman and Woodward, Higher Mathematics, Chap. XI.), Art. 11. Also see F. Cajori, Iistory of IMathematics, pp. 341-347. t " Gauss in 1799 showed that the differential equation meets its limitations very soon, unless complex numbers are introduced." MISCELLANEOUS NOTES. 203 mathematicians saw that any marked advance in this direction was impossible without the aid of new conceptions and new methods. But it was not until a comparatively recent date, that wider regions were discovered and begun to be explored. "A new era began with the foundation of what is now called functiontheory by Cauchy, Riemann, and Weierstrass. The study and classification of functions according to their essential properties, as distinguished from the accidents of their analytical forms, soon led to a complete revolution in the theory of differential equations. It became evident that the real question raised by a differential equation is not whether a solution, assumed to exist, can be expressed by means of known functions, or integrals of known functions, but in the first place whether a given differential equation does really suffice for the definition of a function of the independent variable (or variables), and, if so, what are the characteristic properties of the function thus defined. Few things il the history of mathematics are more remarkable than the developments to which this change of view has given rise." - The leading events in the early history of this new theory are: the publication of the memoir on the properties of functions defined by differential equations, by Briot and Bouquet in the,Jouzrnal de l'Ecole Polytechnique (Cahier 36) in 1856; the paper on the differential equation which satisfies the Gaussian series, by Riemann at Gittingen in 1857; and, perhaps, most important of all, the appearance of the memoirs of Fuchs on the theory of linear differential equations with variable coefficients, in Crelle's Journal (Vols. 66, 68) in 1866 and 1868. t The only work in English which employs the function-theory method in discussing differential equations is that of Professor Craig. $ A knowledge of the theory of substitutions, as well as of functiontheory, is required for reading some of the modern articles on differential equations. * See G. B. Mathews, a review in Nature, Vol. LII. (1895), p. 313. t Albert Briot (1817-1882); Jean Claude Bouquet (1819-1885); Georg Friedrich Bernhard Riemann (1826-1866), the founder of a general theory of functions of a complex variable, and the inventor of the surfaces, known as "Riemann's surfaces"; Lazarus Fuchs (born 1835), professor at Berlin. $ T. Craig, Treatise on Linear Differential Equations (Vol. I., published in 1889). See Note J for the names of other works on the modern theories. 204 DIFFERENTIAL EQUATIONS. Professor Lie * of Leipzig has discovered, and since 1873 has developed, the theory of transformation groups. This theory bears a close analogy to Galois' theory of substitution groups which play so large a part in the treatment of algebraic equations. By means of Lie's theory it can be at once discovered whether or not a differential equation can be solved by quadratures.t A forthcoming work by Professor J. M. Page on differential equations treated from the standpoint of Lie's theory has been announced. t The theory of invariants of linear differential equations is one of the later developments in the study of differential equations. While it plays a very important part in both of the modern theories referred to above, yet, to some extent, it can be studied without a knowledge of these theories. ~ It has been found that differential equations, like algebraic equations, have invariants. An invariant of a linear differential equation is a function of its coefficients and their derivatives, such that, when the dependent variable undergoes any linear transformation, and the independent variable any transformation whatsoever, this function is equal to the saine function of the coefficients of the new equation multipled by a certain power of the derivative of the new independent variable with respect to the old. The introduction of invariants into the study of differential equations is due to E. Laguerre of Paris. i Those who have made the most inm* Sophus Lie was born in Norway and educated in Christiania. He has been Professor of Geormetry at Leipzig since 1886. He has expounded his theory in the following works: Theorie der Transformationgr ppen, Vols. I., II., III. (1888-1893); Vorleslugen iiber contieuierliche Gruppen (1893). See p. 207 for his work on Iifferential Equations. t For an elementary introduction to Lie's theory of transformation groups, and its application to differential equations, see articles by J. M. Page: "Transformation Groups," Annzals of lMathematics, Vol. VIII., No. 4 (1894), pp. 117-133; "Transformation groups applied to ordinary differential equations," Annals of IMathematics, Vol. IX., No. 3 (1895), pp. 59-69. Also see J. M. Brooks, "Lie's Continuous Groups," a review in Bull. Amer. 3ilath. Soc., 2d Series, Vol. I., p. 241. t By The Macmillan Co. ~ See Craig, Linear Differential Equations, pp. 19-22, 463-471; and the memoir of Forsyth referred to below. Il In his memoirs: "On linear differential equations of the third order," Comptes Rendus, Vol. 88 (1879), pp. 116-119; " On some invariants of linear differential equations," Ibid., pp. 224-227. MISCELLANEI'OUS NOTES. 205 portant investigations on these invariants are Halphen and Professor Forsytho Their memoirs* are among the principal sources of information on the subject. NOTE J. Works on Differential Equations. A brief list of books on differential equations may be interesting and useful to those who intend to continue the study of the subject. For convenience these books are divided into three groups. The first two groups consist of works which have been written from the older point of view; the third contains works in which any of the modern theories of functions, substitutions, transformations, invariants, etc., are more or less discussed. The first group is made up of the smaller and more elementary works; the second includes the larger and more advanced. OSBORNE: Examples of Differential Equations with Rules for their Solutions, vii + 50 pp. Boston, 1886. BYERLY: Key to the Solution of Differential Equations, being pp. 296 -339 of his Integral Calculus, edition of 1889. Boston. EDWARDS: Elementary Differential Equations, being Chaps. XIII.-XVII., pp. 211-277 of his Integral Calculus for Beginners. London, 1894. JOHNSON: Differential Equations, being Chap. VII., pp. 303-373, of Merriman and Woodward, Higher Mathematics. New York, 1896. STEGEMANN: Integralrechnung (edited by Kiepert), Chaps. XIII.-XV., pp. 407-563, 6th Aufl. Hannover, 1896. Ist Aufl., 1863. AIRY: Partial Differential Equations, viii + 58 pp. London, 1866. II. DE MORGAN: Differential and Integral Calculus, Chap. XI., pp. 183-215; Chap. XXI., pp. 681-736. London, 1842. * G. H. Halphen (1844-1889) of the Polytechnic School in Paris. "Mémoire sur la réduction des équations différentielles lineaires aux formes intégrables," Mémoireses des avants Étrangers, Vol. 28 (1884), pp. 1-301. Chap. III., pp. 114-176, in this memoir treats of invariants. A. R. Forsyth, " Invariants, Covariants, and Quotient-Derivatives associated with linear differential equations," Phil. Trans. Roy. Soc., Vol. 179 (1888), A, pp. 377-489. 206 DIFFERENTIAL EQUATIONS. PRICE: Infinitesimal Calculus, Vol. II., pp. 513-707. London, 1865. HYMERS: A Treatise on Differential Equations, viii + 180 pp. 2d edition. London, 1858. BOOLE: A Treatise on Differential Equations, xv + 496 pp. London, 1859. The same, new edition with supplementary volume, xi + 235 pp., by I. Todhunter, 1865. FORSYTH: A Treatise on Differential Equations, xvi + 424 pp. London, 1885. JOHNSON: A Treatise on Ordinary and Partial Differential Equations, xii + 368 pp. New York, 1889. MOIGNO: Calcul Intégral, pp. 333-783. Paris, 1844. DUHAMEL: Calcul Infinitésimal, t. II., pp. 122-490. 2d edition. Paris, 1861. SERRET: Calcul Intégral, pp. 343-676. Ist edition. Paris, 1868. HOPEL: Calcul Infinitésimal, t. II. (1879), pp. 287-472; t. III. (1880), pp. 1-237. Paris. LAURENT: Traité d'Analyse, t. V., pp. 1-320; t. VI., pp. 1-223. Paris, 1890. BOUSSINESQ: Cours d'Analyse Infinitésimal, t. II., 1, pp. 177-229; t. II., 2, pp. 1-7, 229-535. Paris, 1890. Du BOIS-REYMOND: Beitrage zur Interpretation der partiellen Differentialgleichungen mit drei Variabeln. Heft I. Die Theorie der 'Characteristiken, xviii + 255 pp. Leipzig, 1864. RIEMANN: Partielle Differentialgleichungen und deren Anwendung auf physikalische Fragen (edited by Hattendorff, 3d edition, xiv + 325 pp.). Braunschweig, 1882. III. FORSYTH: Theory of Differential Equations, Part I. Exact Equations and Pfaff's Problem, xiii + 340 pp. Cambridge, 1890. DEMARTRES: Cours d'Analyse, t. III., pp. 1-134 (in 4~ lith.). Equations differentielles et aux différences partielles. Paris, 1896. KOENIGSBEIIGEI: Lehrbuch der Theorie der Differentialgleichungen mit einer unabhangigen Variabeln, xv + 485 pp. Leipzig, 1889. JORDAN: Cours d'Analyse, t. VII., pp. 1-458. 1st edition 1887, 2d edition 1896. Paris. PICARD: Traité d'Analyse, t. II. (1893), pp. 291-347; t. III. (1896), xiv + 568 pp. Paris. PAINLEVÉ: Leçons sur la Theorie Analytique des Équations Différentielles, 19 + 6 + 589 pp. (Lith.). Paris, 1897. MISCELLANEOUS NOTES. 207 LIE-SCHEFFERS: Vor11SUiigeii liber Differentialglcichungen mit bekannten infinitesmalen Transformationen, xiv + 568 pp. Leipzig, 1891. PAGE: Differential Equations from the Standpoint of Lie's Transformation Groups. (Now in Press.) New York, 1897. GOURSAT: Leçons sur l'integration des équations aux dérivées partielles du premier ordre, 354 pp. Paris, 1891. GOURSAT: Leçons sur l'integration des équations aux dérivées partielles du second ordre, t. I., viii + 226 pp., Paris, 1896; t. II. (en cours d'impression), 1897. POCKELS: Ueber die partielle Differentialgleichung u + K2U - 0 und deren auftreten in der mathematischen Physik, xii + 339 pp. Leipzig, 1891. MANSION-MASER: Theorie der partiellen Differentialgleichungen erster Ordnung, xxi + 489 pp. Berlin, 1892. POINCARÉ: Sur les équations de la physique mathématique, 100 pp. Paris, 1894. PAINLEVÉ: Leçons sur l'intégration des équations différentielles de la mécanique et applications, 4to. (Lith.) 295 pp. Paris, 1895. CRAIG: Treatise on Linear Differential Equations, Vol. I., ix + 516 pp. New York, 1889. * HEFFTER: Einleitung in die Theorie der linearen Differentialgleichungen mit einer unabhlngigen Variabeln, xiv + 258 pp. Leipzig, 1894. KLEIN: Vorlesungen uber die hypergeornetrische Function, 571 pp. (Lith.) Gôttingen, 1894. KLEIN: Vorlesungen iber lineare Differentialgleichungen der zweiten Ordnung, 524 pp. (Lith.) Gôttingen, 1894. t SCHLESINGER: Handbuch der Theorie der linearen Differentialgleichungen, Bd. I., xx + 486 pp. Leipzig, 1895. Bd. II., Th. 1, xviii + 532 pp. 1897. * See review by M. Bôcher, Bill. Amer. Math. Soc., 2d series, Vol. III., p. 86. t See review of Vol. I. by G. B. Mathews, Naturte, Vol. LII., p. 313; and review by Bôcher, Bull. Amer. Math. Soc., 2d series, Vol. III., p. 146. ANSWERS TO THE EXAMPLES. CHAPTER I. (p stands for dy.) dx Art. 3. 2. (X2 -2y2)p2-4pxy - x2 =. 3. (1 +p2)8 =r 2(d1Y' Jdx2 (d) dx Page 11. dx _ dy 1. _ 8.. +2- xy =0. /1 -sx2 - y2 dx2 dx 2. pV'1 - x2 (= y. 9. dy q+ 2y = 0. 3. d 7dY 6 y. 10. p2x a. dx3 dx 4. 12p2y = (8p3 - 27)x. 11. Y2 = 2pxy - 2. 5. y -px +p-2 p 12. d-=0. dx3 6. 8 ap3 27 y. 13. xy J2 d 13. xy^+ } == q- x -- 7. p2(1 - x2) +1 = 0O dx= \.dx dx 14. x2d 2y 2xd dx2 dx CHAPTER II. Art. 8. 2. y-/1 — x + x- 4- - = c. 3. y — = c(a + x) (1 - ay). 4. tanl c(l - ex)3. p 209 210 ANS WERS. Art. 9. 2. xy2 c2(X + 2 y). 3. y = ce3Y. 4. c(2 2 + 2xy- 2)23( + )+ 2y (Y3-1l)x+2y Art, 10. 1. (y-x+ 1)2(y+x- 1)5=c. 2. (x+ 2) (y2 y -6) 2+l =. Art. 13. Y3 3. a2x + - x2 - x2y = c. 4. ax2 + bxy + cy2 + gx + ey = c 5. X2y2 + 4 x3y- 4 y3 + y3-xey+e2 + x4 = c. Art. 16. e3 3. 2 a logx + a logy-y = c. 4. x2e + my2 = cx2. 5. 2 + - =c. y Art. 17. x Y3 1 1. +log -=c. 3. 21ogx-logy -+c. y x2 xy Art. 18. 1. e(x2 + y2)=c. 2. x2 -y2=cx. 3. y3 + 2=cy. 2x 4. xy + 2+ = c. y2 Art. 19. 36 24 10 _15 3 2. 5x-r13I-12 x3y+ = c. 3. 6 Vxy - x2 =C. Art. 20. 2. y=(x+c)e-X. 3. y = tan x-1 + ce-tan". 4. y=(e + c)(x + )". 5. 3(2 +l)y = 4x3+ c. Art. 21. 2 x2 3. 7 y-3 = cx3 -3 x3. 4. y = c(l - 2) 1-X 5. y3 = ax + cx /1 - x2. ANS WERS. 211 Page 29. 1. + y tn Y+c 4. tanxtany=k2. *. =- =tan y --- a a x y+x 2. x2=2cy+c2. 5. log ---Y =c 3. Y =j X2 e-^ 6 )6. y = x(1 + ce). 7. 60y3(x + 1)2=106 + 24x5 15x4 + c. 8. x2-xy+y2 +x -y=c. x 9. y =ce V-x2+4 x_ a X /l -x2 10. cx=eY. e 11. X -y +2 2y2 —2a22 -2 b2y2=. 12. y(x2 + 1)2 = tan-1x + c. tan 16. logV x2 -+y2 V+m tan-y =c. X3 X 13. log cy = - 3y _ 17. r=cem0. 14. y = ax + cxYl/ - x2. 15. 2 - y2 -1 =cx. 18. x +yeY= c. 19. y-n+l = ce(n-1)sinx + 2 sirl x + n-i 20. (x +1)ey=X2+2x+c. 23. y + 2y3 +-2y2 C. 21. =x2++y22. 21. =2 + x +-ce22 224.. y-2 a2y2+ 4-2 xa22+2 2y2c. d2- - ayn+2 y2 _ _2 y 22. + c. 25. -— "2 f y2 + ce ~Y. n+-2 x x 26. y2=a2 +c2 27. (x+ Va2 +)y = a2 log( + /a2+x2)+ c. 28. log Vx y2 + tan-1 Y = c. 29. (4 b2 + l)y2 2 a(sin x + 2 b cosx)+ ce-2b. 30. x2 +y2 = y. 1 32. c(y- b)= — b 31. 3 y22 x2 e = c2. 1 + b 33. 9 log(3 y + 2x + -2-)= 14(3 y - 3 x + c). 34. x2y -2 xy log cy = 1. 212 ANSWVERS. 4 1237. ax C. 35. 4 x2y = + cxy-r. 37x x c. y 38. alogX-y+a-y=. 36. cy = ex. 2 x- y - a CHAPTER III. Art. 22. 3. = c, x +- y = c, xy -+ x2 + y2z = c. 5. x2 + 2 y2 c, x3 + 2 = c. 4. 289(y + c)3 = 27 ax7. 6. y - 4x + c, y = 3 x + c. Art. 24. 2. x =- {(1 + p2)- log (p - 1) - tan-1lp, with the given relation. 22 3. log (p- x)= x + c, with the given relation. p-x 4. 2y=cx2+-aArt. 25. 1. y=c-[p2 + 2p+21og(- 1)], x =c-[2p+21og(p-1)]. 2. y =c-alog(p -1), xc + a log p~ p -l 3. y2=- 2 C-. Art. 26..1. x= logp2 + 6p + c. 2. y!l-x 2. y - c = x —x2-tan-11 —x 3. 2 y4+ c = a(p +- 1 + p2)- log(p + l/1 + p2, x = aVi + p2. 4. x c = a log(p + /1 +p2), y = av + p2. Art. 27. 2 4 1. y2 + cyx5-1 = c2x5-1. 2. y2 = 2 c + c2. Art. 28. 3. y= c+ sin-c. 4. e = ce2 + c2. 5. y2 =x2 +1+ c. Page 38. i. sin-1 Y-= log cx. 2. y-=-c(x-b) +-a 1. sin-1 y- =logc ex. 2. y = c(x - b) + - x C ANSWERS. 213 3. (x2 - y2 + )(X2 _y2+CX4) = O. 5. (y -6 + c)(y-3x +c)= O. 4. xy = c c2x. 6. ac2 + c(2x - b)- y2 = 0. 7. 0 + -- p(r2)dr, where 0 tan-1 Y and r2 = x2 + y2 rJ r 2 - 02 (12) X( x 8. tan-1 y + c = vers-12 a /x2 + y2. x 9. sin-1 y = x/x2 - I - c. 10. x2+y2 _4cx 3 e2=. (Put x2 - 3 y2 = v2.) 11. 2 + 2 2 2c(x + y)+ c2=0. ],-1 ch2 -_____ 1++!)lc 14. y2 - CX2 + c- 0. 12. y + -/y2 + nx^2 = ex n C + 1 13. y2- b = (x + c)2. 15. y( l cos x)= c. 16. y - siln-1 - c y - cos-l 1- - x- -- ) = 0. 17. (y + c)2 + (x - a)2 = 1. 19. (y - cx2) (y2 + 3 x2 - c) = 0. 18. y= 2 cV + f(c2). 20. y/=c(x-c)2. X2 21. (X3 -3y+c)(e2+cy)(Xy+ Cy + 1)=0. 22. ax + c -- -p2 - mp + m2 log (p + n), with the given relation. 23. ey = ce + c3. 25. (x + c)2 + (y - b)2 = 1. 24. log Y =c+y. 26. y =cx +m. x + V2 -y2 c 27. y2 CX+ C3. CHAPTER IV. Art. 33. 2. y = cx + c2, 2 + 4 y = 0. 4. x2(y2 - 4 3) =0. 3. (y+ x-c)2 =4xy, xy=O. 5. (y-x+c)3=a(x+y)2, x+y=O0 Page 49. 1. 2y = c2 a, y2 = ax2. 2. a3x + cxy + c2 0, singular solution is x(xy2-4 a3)=-0. x-( is also a tac-locus. 3. (y - e)2 = m2 -+ c2, y2 + m2x2 = m2 214 ANSWERS. 4. y = cx + /b2 + a2c2, b2x + a2y2 = a2b2. 5. y -cx -c2, x2 = 4 y. 6. y2 = 4 a(x-b); 1 +4x2y-0, x = 0 is a tac-locus; 27y =4x3; y2 = 4 mx. 7. (y + c)2 = xa; x = O is a cusp locus; there is no singular solution. 8. (y + c)2 = X(X - 1)(x - 2); singular solutions are x = 0, x = 1, x = 2; x = 1 are tac-loci. The curve when c = 0 consists of an oval cutting the axis of x at the origin and at x = 1, and a curve resembling a parabola in shape, having its vertex at the point for which x = 2. 9. x{x3 -c(y + c)2} = 0; singular solution is 4 y3 +27 x3 = 0; =0 is a part of the general solution, and is the cusp locus for one part of the general solution and the envelope locus for the other part. 10. y = cx + /a2c2 + b2, b2x2 + a2y2 = a2b2. 11. x2+ y2 - c(x2 - y2)-1 + c2 = 0; that is, + = 1. Singu1 + c 1 - c lar solution is 4 4-2 xy2+- y4 - 4 x2 - 4y2 + 4 =0; that is, (x + y ~v)(x + y -+ () (x - y + /2)(x - y -2) = 0. The general solution is the system of conics touching these four lines. CHAPTER V. Art. 43. 3. ny= x + c. 4. y = /ax-x2 + a vers-1 2 x 2 a 5. r = c- K cos 0; when c = K, the cardioid r = K(1 - cos 0). 6. cr = eK. Art. 47. 4 4 4 3. The ellipses 2 x2 + y2 = c2. 4. y3 - x = 3. 2c 5. The confocal and coaxal parabolas r = 1 - cos 0 25 y 6. sec 50 + tan 50 = cer. 7. r= ceZcot0, r = x2+ y2, 0 = tan-1 Art. 48. 2. s = at2 + ct + c2, s at2. 3. 2 s = V2-g t + c, s gt2. Page 60. 1. y =ce='. 2. x- + y3 =a 3. 3 y2 = 2 K (X3 ). ANSWERS. 215 4. s = K sin 0, the intrinsic equation of a cycloid referred to its vertex, the radius of the generating circle being Y K. I 1 5. The lines r sin (0 + a)= -, and their envelope the circle r = - K K 6. The parallel lines (m sin a - n cos a)x - (m cos a + n sin a)y = c sin a. 7. x2 + 22 = c. 8. The system of circles passing through the given point and having their centres in the given line. 9. x2 + y2 = 2 a2 log x + c. 10. x2 -y2 =c2. 11. rn = cn sin n0; r2 = c2 sin 2 0, a series of lemniscates having their axis at an angle of 45~ to that of the given system. 12. r2 - K2 = cr cosec 0. 13. r = evc'-2. 14. Parabola (y - x)2 - 2 a(y + x) + a2 = 0. 15. x2+y2- =a2. 16. x2 +y2 = 2 cx. 17. log c (y + v2 - x2) = Y (Y + /2 ). y2 18. x2 = c'2e. 19. r = c sinn; r = c sin; r= c(1- cos ). 20. r =ceO. 1-o20 21. (y-x)V/2=c + v 22. re v = c. /Y - ax 23. r a(l - e2) 24. r = c(l + cos 0). 1 - e sin (0 + c) 27. The ellipses that have the fixed points for foci. 28. The ellipses that have the fixed points for foci. 29. The ellipse K2X2 + a2y2 = a2K2. 30. The hyperbola 2 xy = a2. 31. The parabola a2x2 = K2(2 ay + K2). 32. The catenary y = a cosh -. (See Johnson, Dif. Eq., Art. 70.) a 33. 4ay + c = 2axa\/4 a2x2 - 1 - log (2ax + a/4 a2x2 - 1). 34. (a) i = ce + e L eL f(t)dt. L (b) i = ce L If i = Iwhen t-, i=Ie. R R (c) i = ce - +. If i = 0 whent = 0, i = - - e-Zt () ji. 216 ANSWERS. -Rt E (d) i = ce L + 2 + L2 (R sin wt - Lw cos wt). (e) i = ce L+ E1 (R sin wt - Lw cos wt) 12+ L2'2 + 12 2b22R sin (bwt + 0) - Lbw cos (bwt + O)}. i2 + -L 2-w2 t 35. (a) i=er Ref (t)dt + cR. t t (b) i = Ce ~. (c) i = ce R. (d) i = ce" c+ 1CEc (cos wt + R Cw sin wt). (d) i- Co2 t -- t t 36. (a) q e -- eCf(t)dt+ ce R. t (b) q = Qe IW, where Q is the charge at time t 0. (c) q = CE + ce R. (d) q = cR+ 1 + 22 (sin wt + Cw cos ot). V2 KVo2t + 1 - 1 37. s — KVO CHAPTER VI. Art. 50. 3. x = cie2t + c2e-4. 4. xc = ce + cae 3 Art. 51. 1. y = e2 (c1 + c2x) + c3e —.. y= e-x(cl + C2X + C3X2) + c4e4x. Art. 52. 3. y = ex(cl + c2x)sin x + ex(c3 + c4x)cos x. Art. 58. 2. y = cie + c2e-x - 2 - 5 x. 3. y = e-x(ci + cx) +e. 4. y= e'2(cl + c2x) + cae-s + e-2 ( Se —5 (e3(x)3. ANSWERS. 217 Art. 60. 3. y= e ~(ccos- - x + c2 sin + e(c3 + 3 x) + } ex — 1. 4. y = —e(ax + b)+ 4e. Art. 61. 3. y = ce-2x + e-(c2 cos x3 x + 3 sin 3 x)+ i(x4 - x 1). Art. 62. 3. y = cle2x + c2e- 2x - i sin 1 x. 4. y =cle- + e2 (lCOS V3 - + c2 sin-3 X sin 3 x + 27 cos 3 x 1 sin x - cos x 730 2 4 Art. 63. 2. y = cle-x + c2e-2 + 1e0 (11 sinx - 7 cos ). e3x ex 3. y = ccos(V/2x + a)+ -(11X2 - 12X -t 50)+ -(4sin2x - cos2x). 121 17 Art. 64. 2. y = C(cos 2 X + a) + x sinx - cosx. 3. y = ciex + c2e- + x sin x + l cosx(1 - 2). Page 80. 1. y = (clee + c2e-) cos x + (cs3e + c4e-~) sin x. 2. y = cle-x cos (x + a) + c2e3x cos (x + f3) + c3e-4x. x3 3 x2 3. y = cl + e-x (c2 + C3a)+ x- + 4x. 3 2 4. y = clcos2x+ c2sin2x+ }(ex - sin 3 x)+ (2x2- 1). 5. y = cie2x + c2e3x + l(6x + 5)+ em - m' - 5 m + 6 ep eenx m 6. y= cea + c2ea- + -e..+ e 8~n7. y c e-2 a 2 a 7. y - cle-2x + C2e4î + Csex + -(X + ). 218 ANSWERS. 8. y = c + c2X + e C si +C4cos +2ax + 6 bx2-1 ax3 +-L (a-3b)x44 — bx5. 9. y = clex + c2e3x + C3e4x + -1(X + -) 10. y csin (nx + a) (ax + b)+ (n2 - x3Sh six9X2- x4 11. y csin (x + a)(ax + b)+ + cosX. 12 48 12. y = cl cos (ax + a) + x sin ax cos ax log cos ax a a2 e3x 13. y =(c+ c2x)ex +e (2x2 - 4x +3). 8 14. y=ccos(nx+a) eex X44 x2(2 x + 3) + 24(2 x2 + 4x + 1)j 14. y-ccos(nx+a)+~ - n 1 (n2-l )2 2+1 (n2+ 1Y 15. y = clear + c2e-" + C3 sin (ax + a) - a-4x4 - 24 a-8. 16. y = cl + c2x + ex(c3 + c4x) + 2 + 1 x. 17. y = clex + c2e-x + C3 sin (x + a) - ex cos x. 18. y= e 2(clcos x + 2s sin 3x) - - (2cos2x + 3 sin2x). 19. y = cie-x + c.-2 + c e + 3e3 2 (x + - 12 \ 12/ 20. y = cex sin (3 x + a) + - ex cosx. 21. y = e-x(cl + c2x + C3x2) + - x3e-. 22. y = e2x(cl + c2x)+ e-x(c3 + C4x) + i ex(x2 + 2 + 7). 23. y/ = ex(cl + C2x + c3x2 + X3 + 1 X4). 24. y = ciex + cle-x - (x sin x + cos x) + 1 - xe2(2 x2 - 3 x + 9). 25. y = cle33 + c2eC - - (2 sin 3x + cos 3 x)- I(sin 2 x + cos 2 x). 26. y = ex(cl + c2 cos x + cs sin x) + xex + -(cos x + 3 sin x). 27. y= cie5 + c2e4 +x+ +0. 28. y = e2x(c + C2X) + c3e-x + À e3. 29. y= cle-x + e2(c2 cos 2x + sin - 2 2(s - aI sin + 3 cos -- ANSWERS. 219 CHAPTER VII. Art. 65. 2. y=cx cos (- logx+a + 2. Art. 66. 3. y = (c + c2 log x)sin log x + (c3 + c4 log x)cos log x. Art. 69. 2. y = -2(c1 c2ogx)+ - 3. y=c +cX - ++4 -Art. 71. 1. y = ci(5 + 2 X)2 + C2(5 + 2 x). 3/3 2. y=(2x- 1)[c + c2(2x- 1) 2 + c3(2- 1) 2 ]. Page 91. 5 221 -21 1. y= C1X2 + X2 2 2 + C3 2 - 2. y = clx- + c2x;-2 + x-2ex. 5 3. y = x2[c1 + C log x + c3(log x)2]. 4. y = cl(x + a)2 + c2(x + a)3 + 3 x 2a. 5. y = -2(-C + c2 logx)+ C3x. 6. y = x(c cos log x + ca sin log x + 5)+ x-l(c3 + 2 logx). 7. y=[cl+_2log(+1)]/+_C34 C _41og(X 1) -+ 52zX 51 V' 41 -225 8. y = clx + c2x-1 + 2 m2 -1 xm 9. y = X2(Ci + c log x) +(m - 2)2 10. y = C.F. of Ex. 3, Art. 66, - (log x)2 2 log x- 3. 11. y = x(ci + c log x) + cX-1 - -1 logx. 12. y =1 log x + ci log x + c2) ûî \ x - 1 220 ANSWERS. 13. y = xm(cl sin log xt + c2 cos log x1 + log x). 14. y = x2(clx3 + c2x-/3)+ 6 + 6o (5 sin log x + 6 cos log x) + 721 (27 sin log x + 191 cos log x). 3721 x CHAPTER VIII. Art. 75. 3. y = xe-2x clx- 2edx + 3 J. 4. e2sy - Se2xx3dx + cl fe26 dx + c2. 4x'2 y 1 4x C4, dx 5. e =e3 x+ Cl ie3 - + c2. in 5 J X2 6. dy = x2y2 + dx. 7. xy2 + c2X5 = c2. Art. 76. | 7xmf+n 2. y = ci + c2x + C3X2 +...+ cn-1 + 1m + n 3. y = cl + C2X + C2 + C4X3 + ~ X2 log x. 4. y = ci + c2x +(6 - 2) sinx-4xcosx. Art. 77. 2. 3 x= 2 al(y - 2 c)(y2 + ci)2 +c2. 3. x/c y2 + y - log (c- + / 1 + cly) = acl /2 x + c2. 4. ax = log (y + /y2 + c1) + C2, or y = cl'ea + c2le-a. Art. 78. 2. 2 ( - b) =e —a+ e-(x-a). 3. y = clx + (c12 + 1) log (x - c) c2. 4. 15 Cly = 4 (x + c12a2)} + C2X + C3. Art. 79. 1. e-a = clx + c2. 3. log y = clex + c2e —x. 2. y2 = x2 + C1X + c2. 4. sin (cl - 2/ y)= c2e-2. Art. 80. 1. y = cl sin ax + c2 cos ax + C +X C4. 2ea 2. y = cleTm + C2e-mx + C3 - C4X -+ Cg52 +a3 (a2 - m2) ANSTWERS. 221 5 i 4a2 1-4a2 3. y=ci + C2x + x2(ex 22 C4x 2 ) when a <, y = c î +C2 + C3 COS ( 4a log cx) when a > 2. 2 C4 Art. 81. $ X 2. 2 acly=cl2ea + e a + C2. 3. y = cl logx + c2. 4. 15y = 8(x + ci)i 2+ C2x + C3. Art. 82. 3. y=A (-2 X2 2(2.2+4) XI _ 2(2 1 2+4)(2.3 4+6)6...) 32 (4 6 _ 4, 3 ' +3(2'2+3+5)x? 3(2'2'3+5)(2'4'5+7) x+... ) +B(x3 15!+ 4. y 2=Ax 1-25 2.4.5.9 2.4.6.5.9.13+) +1 B X2 X4 X6 +B.3 x47 2+2 3+4 3.7+4 711 + 2 4 6+ X 1.3 1.33.3-7 1.3.5.3.7.11 5. y-A (1-n.n+- 12-+n n-2.n+ n + 3.-... + B(x-n — n +23+ n -1. n-3. n +2. n +4 *L3 15 \ 1 y= Ax1 - nn-1 - n 2 +n l - 1n. - 2. n - 3 4 y=Az1-2.2n_-1 2.4.2n-1.2n- 3 +Bx-n- (l1n +n+-1 ~ +2 z-2 2+ 1n? + - 2 n + 3. 7 +4 x+...) 2.2n+3 2.4.2n+3.2 n+5 Page 107. 1 y2y d+2 xy = c2 4. x3 + X2 + 3y =X2 C. x d2+ dy 2. 2 ay + x2 = cl/a2 - x2 + c2. 5. y = c + 2 + Ceax + c4e-X. 3. (1+x+x2)y=cx2+C2x+ cg. 6. cly = c2eCx - nVl+a2 c2. 7. xy/2 - 1 = sec-1 x + CiX;2 - 1 + c2 log (x + x2 - 1)+ Ca. 8. y = c2 - sin-1 c1e-. 9. y = ci sin-l x + (sin-1 x)2 + ca. 222 ANSIWERS. 10. y = ci sin2 x + c2 cos x - c2 sin2x log tan - 2 11. y = c2e((x -)+ cil + cil xex -îe-(dx)2 + C3. 12. y= ax log x + clx + c2. y=c + 3+ S 2 14. y:cix2+c2x++ C3+ ---? 1 12 16 13. log y - 1 = clx + c2 15. y- cle-x + C2 + ex 16. y = e-sinx esin(clx + c2)dx + cae-sinx - si 1. 2 17. y = cl(l - xcotx)+ c2 cot x. 18. a log(y + b)= x + c. 19. (clx + c2)2 + a = ciy2. 20. d f (x)dx + Cl, xd-d =i af(x)dx + C2, -d 2xdy +, 2 y x2f(x)dx + cs. x2 - 2 y =dx 21. y-b = - log sec aK(x - c). CHAPTER IX. Art. 87. 2. y= Ax + B X2ei3 dcx + Sx-2e Sx3e (dx)2. 3. +y =Ae4+ BCe(4x3-42x2 150-183). 4. y = c ir(x+ 4 ) Art. 88. 2 2 2 2 3. y ce2+ c2e2x e3 dx. 4. y ciex + c X2e2. Art. 91. 2. y = ce 8x sin ( logx +a). 3. y = (cl sin /6 x + c, cos /6 x) cos x. 4. y = ex(cix.2 + c2x). Art. 92. 2. y =csin l21ogttan4aV 3. y=ci sin (X2 + a)+-. 4 =y=cs(S+ogtan ) Sil 2 4 4. y=ccos -a a 1 4. y-l cos 2c2 2x2 -a2n,2 X2 xX2 a2X2 ANSWERS. 223 Page 120. 1. y =l(clenx c2e-nx). 4. yx/ + x2 = cl log (x + 1 +x2)+ c2. 2. xy = c sin (nx + a). 5. y = clex + c2e3(4 X - 42 x2 + 150 x- 183). I bx2 3. y cx sin (x + a). 6. y ceb2 sin (x/b + a). 7. y = e-x2(clex2 + c2e-xv). 8. y= ex (cl log x + c2). 9. y = cl+ c2(x sin-lx +V1 -x2) — x(1 -x2)-. 10. y = clx + C2 cos x. 11. y = clX2 + c2x + c3 (x2x-3e-xdx-x ix -2exd). 12. y =clea sin-lx + c2e-asin-x. x2 x'2 3c2 13. y= Ci + c2xix-e 2dx + Sx x-2e 2 zxe2f(x)(dx)2. 14. 2 y = x(c1e2" + c2 - x). 17. y2O = CX2 + ClX. 15. y,= c sin a - X2 a). 18. y" +(x - c)2 = k2 a 19. c co s lVx 16. y = ci sin (nx2 - 1 + a) 19. y \ X c ) CHAPTER X. Page 124. 1. The circle of radius 1 K Z+C' C+C' i x+c' c 2. Acatenary, y = e c + e ) 3. y2 + (x - a)2 = c2, circles whose centres are on the x-axis. 4. (x - a)2 = 4 c(y - c), a system of parabolas whose axes are parallel to the axis of y. 5. x + cl = c vers-' - /2 cy - y2, the cycloid obtained by rolling any c circle along the x-axis from any point. 6. The ellipses a2y2 + K2(X - c)2= a4, if the normal is - K2 times the radius of curvature. The hyperbolas a2yJ2 - K2(X - c)2 = a4, if the normal is + K2 times the radius of curvature. A set of parabolas if no constant is introduced at the first integration. 224 ANSWERS. 7. The elastic curve represented by the equation {4 K2 - (x2 - a2)2}~ dy = (x2 - a2)dx. 8. s = cleKt + c2e-Kt, when accel = K2 distance from the fixed point. s = Cl sin(Kt + c2), when accel = -- K2 distance from the fixed point. 9. s=- at2. 10. The relation between time of motion and the distance passed over is t = c + /cVs2 - cs + c log(/s + /s - c)}, according as the ac2K 2 celeration is Ks2 a n2 11. s = cosh nt, if the resistance of the air is - times the square of the velocity. /2 Kv20t + I - 1 12. s =, if the acceleration is - K times the cube of the KVo velocity. 13. T = 2 7r;T/. (Emtage, Mathenmatical Theory of Electricity and Magntetism, p. 85.) 14. s= 1 + (so - l) cos Kt, v =- K(So - l) sin Kt, where K =\(J.,e HINT: Put s - I equal to a new variable. 16. 2, if acceleration is -- t t t 17. i 7 e-elf - e-ve2f'(t)dt \,fR2 C2 - 4 L C. t - t +cle i + c2e T2, where T = 2LC and T = 2L RC- V/2C'2 - LC RC + /RC'2 - 4 LC 18. Same as in 17 with f(t) substituted for f(t). 19. i = e 2L(Ci + c2t). 20. i = Isin 21. - a = cle —tK sin (V2-K2t+C2) for O> K; 0 - a = le-(K-vK2-W2)tî~ c2e-(K+/K2 -2)t for o < K. 22. 6 EIy = P(3 12x - x3). 23. 24 EIy = w(4 1x - x4). ANSWERS. 225 24. The general solution is +I W2 WE ++ y = A cosh -Qi x + B sinhQx+ 2 + On applying the conditions of Ex. 22 to determine the constants, A I= 2, B= O; Q2 and therefore, y= E2 - cosl x)+ w2. CHAPTER XI. Art. 98. 2. x=e6t(A cos t + B sint), y = e6t(A - B) cost + (A + B) sint]. 3. x = cle-51 + c2et + f et - 5- t - 23-, y - Ce-5t + c2et + 4 et - 3 t - 1 4. x - cle-t -+ 2e-6t +J-39 t_ — 6 —2-et, y = -le-t +q4 c2e-6t — y t+5- + -42 4et. 5. x = (ci+c2t)et + (ca+ c4t)e-t, 2 y=(c2-cil-c2t)et- (c3+c4+c4t)e-t. Art. 99. 3. x2 = 2 + Cl, X3 y3 + c2. 4. = y log +, y - x =cxy. 1 - X X 3 5. 3 - y3 - C13? X3 + _ C2. 6. axI2 + by2 + cz2 = cl, a2x2 + b2y2 + c2z2 = c2. Art. 103. 3. (y+z)e =c. 4. x-cy-ylogz=O. 5. ex2(y + z2 + ) = c. 6. y(x + z)= c(y + c). Page 143. 1. x - Cie-st + 1 et -.- et, x +- = c2e-t +e2 - -- et. 8 1 6 2. x (c sin t + c2 cos t)e-4t +31-93 26 17 3, -4t - 2 et. y = [(c2 - ci) sin t - (c2 + c1) cos t]e-4t - + 13 17 3. y = (cl + c2x)ex +-3c 3e-2- X, z = 2(3c2 - ci -- c2x)e - c3e- - 4. x = cie-2t + c2e-7t - + 5 t- iet, y = - iJ ce-2t + c2e-7t +- -9- t +- et. Q 226 ANSWERS. 5. =+ a, x+y=bet. 8. xy+yz+z=c(x +y+z). 9. logxyz+x+y+z=c. 6. x2+y2+z2=c, x2-y'2-z2=c2. 6. Y7 y2 l +2y 2 z 10. (y + z) (u + c)+ z(x-u)=O. 7 Y+ZZ+X x y 11. X2 +XY2-u + X2 = C. 12. 2 + (x - a)2 + (y - b)2 = h2. x = cl cos v3 t + c2 sin /3 t + c cos V2 t+ c4 sin v2 t + 5 te3t 1 cos 2 t, 1 y = -3 cl cos /3 t - 3 c2 sin v/3 t - 2 Cg cos V2 t - 2 c4 sin v2 t + 6 te3t + + 12 97 e3t. mt 14. x =e c - cos mt + c sin ) + e cs CS + c4 sin m. rnt mt y-e,(cl sin mt m- cl co - ( mt mCOS C y eVi (cîsin~L-c2cos m ~e (c4cos1- e c cs~ 3si-rc)i ( /2 V/22 - v2 c2 ) x = ai sin Kt + a2 cos Kt+ a3, 15. y = b sin Kt + b2 cos Kt + b3, z = cl sin Kt + C2 cos Kt + Cg, where K2 = 12 + m2 + n2; and the arbitrary constants are connected by the following relations: mci - nb, _ nal - Ici _ bl - mai k a2 b2 C2 lal + mb + ncl =, a= b3 =c. I m n 16. See Forsyth, Diff. Eq., Ex. 3, Art. 174; Johnson, Diff. Eq., Art. 242. 17. x + miy = cle(a+ma')t + c2e-(a+mla')~t, x + m2y = C3e(a+m2a')it + C4e-(a+m2a')tt, where mi and m2 are the roots of a'm2 + (a - b)m - b = 0. Ex. 16, p. 269, Johnson, Diff. Eq.; Ex. 4, p. 270, Forsyth, Diff. Eq. 18. When the horizontal and vertical lines through the starting point in the plane of motion are taken for the x and y axes, the equation of the path is x = vot cos 0, y = vot sin - gt2; X2 and the elimination of t gives the parabola y=x tan (- ~ gvo2. vo2 COS2 0 19. Axes being chosen as in Ex. 18, x = cos 0(1 - e-ct), y = - t + CVo sin + g (1 - e-nt). C C C2 ANSWERS. 227 20. For upper sign: the hyperbola (aly-bx) (b2x-a2y) = (alb2-a2bi)2. For lower sign: the ellipse (aly —blx)2 + (a2y-b2x)2 = (alb2-a2b)2. CHAPTER XII. Art. 108. 2. z=-px+qy+pq. 3. p=q. 4. q=2yp2. 5. = pq. 6. ^xzaa2+ -2 - zx = 1 or yz ya z- ax2 Va/ ôx ay2 \y/ Oy Art. 109. 2. yp - xq = 0. 3. (I + np)y + z(lq - tp)=(m + nq)x. 4. X2O0u 82z a2O 0 4.-+ y2t+yO = 2 5. 5 =a22z Ox +y y2 -X2 y 2, z = ea(x- y). 3. Ix + my + nz = 0(x2 + y2 + z2). 4. -f —1~ 5. f(x2 - Z2, X3 - y)= O. x y \x Z Art. 117. 2. xyz-3u=+ (Y) Art. 119. 5. z = ax + y + b. a Art. 120. 2. z = ax + by + 2vab. Art. 121. 2a2 2. z4a2+1-1 = bxya. 3. (z + a2)3 = (x + ay + c)2. 4. 4c(z-a)= (x+cy+ b)2+4. 5. z2 [z/z2-4 a2-4 a2 log(z + Vz2-4a2)]=4(x+ ay+ b). 228 ANSW ERS. Art. 122. 2. z-=(x +a) +(y+a) 2+b. 4. z = (2x-a)3 +a2y + b. 3. z =ax + a2y2 + b. 5. z - -( a + a) + - + b. 6. z2 = x-/x2 2 -+ y/y2 2 + 2 log x + x2 + a2 b. = _ lY- +b. Y -/+y2-_a2 Art. 123. 2. z = (x + a)(y +b), S.I. is z = 0. Another forin of the C.I. is 2V /z - + ay + b. 3. C.I. is z = a2y2 + (ax + b)2. Art. 12 X3y 3. z = y + x (y) + (y). 4. z =F(y)x-+ f(y). Art. 126. 2. x-f(z)+ (y). 3. z = (L)+ F(). Art. 128. 2. z = 0(y-2x)+ (y - x). 3. z = 0(y +3x)+ (y -2x). Art. 129. 1. z = x2O1(y + X)+ X02(y + X) + 3(y + x). Art. 130. 3. 4. Y + 2. 12 6 Art. 131. 4. z= -(x y) +e3(y - x). 5. z = exz(y) + e-x(x + y). 6. z = ex(y - x) + e ---2/(y + x). Art. 132. 3. — (yex+2y + 18ia X3 + X2?j + X2 + Xy - X), sin(x 2y)- xeY, - ie"-Y + i X2y + 3 Z2 + 1 Xy + * X + 4y + *1. ANS WERS. 229 Art. 133. 2. (x`2y) + x (x2y) + X 30 3. z = (x2 + y2) + (x2 - y2). (Put x for l X2, y for ~ y2.) Page 187. 1. z =x0(- 5. (z-Y) x- y = (- ) '\ yx — 2. x2 + y2 + z2 =,z - ) 6. alogz = ax + (1 - ac)y + b. 11. 1~ tan (x~ y)eY p(zVeot x ~ y 3. (a-l)z + xC= Ysec ( x + y)3 - (2 2 912. - y -x =x g 10. x + y + z = (xy). 2/1. 1 + tan (x ) e 2x + y) sec (x + y) 2 12. -l + z2-log =x + ay + b. z 13. xz = ay 2x/ b. 15. zax+ a 15. - = axq + —? + c. 14. (z-b-alogx)2 = 4ay. n + /n2 - a2 16. z =ax+ by+cV/l+a2+b2. S.I.isx2 +y2+z2= c2. 17. z = ax +(1 -/a)2y + b. 18. z = axey + 1 a2e2y + b. 19. az - 1 = cetxay. 20. (1) 2z=(j+ay? + b; (2) z=xy+y/x2 - a12 + bi; (3) z = xy + xv/y2 + a22 + b2. 21. z = ~ a log (x2 + y2) + /1 - a2 tan-l- + b. (Change to polar co-ordinates.) 21 _-.~1 =xm+x ylt+ 22. 2- zl- = +1 2 - I + 1 in +1 23. z=a x++ /1-a2x/x-y+ b. (Put /x+y =u, /x-y = -.) 24. z = axy + a(x + y) + b. (Put xy = v, x + y = w.) 230 ANS WERS. 25. 1 + az - 2=/x + ay + b. 27. zy2 = xy F(x)+- (y). 26. z-=x3y2 + (y)logx + /(y). 28. y = x (z)-+ (z). 29. x =(z)+~ (x+y +z). 30. (n- 1)zy + ax= e(?1-l)"x(y).f(y). 31. z= f e-f(X)adxL ef()aF(y) dx + 0O(y)dx + (y). 32. z = 1 yx2 log x + x2p(y) + (y). 33. z = i x2y - xy + P (y) + e-"x(y). 34. 4z = x2yJ2 + 0(y) + /(x). 24 C 35. z=a - + r+F(y —b)+f(y-ax). 36. y + xq(ax + by + cz)= i(ax + by+ cz). 37. 3z = ax(y2- 1)(y2 + 2)+ i(x) x/1 y2 + /(y). 38. 4 z =5 32+f(Y)-dx + k(Yy). x INDEX 0F NAMES. (The numbers refer to pages.) Airy, 205. Gauss, 107, 202. Aldis, 150, 153. Gilbert, 193. Glaisher, 48, 106. Bedell, 145. Goursat, 207. Bernoulli, 28. Gray, 106. Bessell, 105, 106. Bôcher, 207. Halphen, 204, 205. Boole, 23, 66, 206. Heffter, 207. Boussinesq, 206. Hilbert, 190, 194. Briot and Bouquet, 193, 194, 203. Hill, 49. Brooks, 204. Hoiel, 206. Byerly, 105, 106, 169, 184, 205. Hymers, 206. Cajori, 202. Johnson, 25, 48, 80, 105, 106, 107,111, Cauchy, 193, 194, 203. 138, 149, 168, 176, 180, 205, 206. Cayley, 40, 48. Jordan, 206. Charpit, 166. Chrystal, 49. Klein, 207. Clairaut, 36, 40, 44. Koenigsberger, 193, 206. Craig, 202, 203, 204, 207. Kowalevsky, 194. D'Alembert, 146, 173. Lagrange, 40, 114, 151, 154, 155, 166. De Martres, 206. Laguerre, 204. De Morgan, 205. Lamb, 184. Du Bois-Reymond, 206. Laplace, 105, 182, 184. Duhamel, 206. Laurent, 206. Legendre, 90, 105, 184. Edwards, 4, 5, 48, 52, 54, 149, 183, 184, Leibniz, 27, 40. 205. Lie, 202, 204, 207. Emtage, 126, 183, 185. Liouville, 200. Euler, 64, 107, 146. Lipschitz, 193. Fiske, 193. Mansion, 193, 207. Forsyth, 48, 69, 80, 86, 94,101,106,107, Mathews, 106, 203, 207. 111, 116, 138, 159, 168, 172, 202, McMahon, 54, 65, 196, 197. 205, 206. Méray, 193. Fuchs, 203. Merriman, 126, 127. Merriman and Woodward, 54,127,176, Galois, 204. 202. 231 232 INDEX 0F NAMES. Moigno, 25, 193, 206. Riccati, 105, 106. Monge, 171. Riemann, 203,206. Newton, 27. Schlesinger, 207. Serret, 206. Osborne, 205. Smith, C., 150, 153, 160. Smith, D. E., 202. Page, 204, 207. Stegemann, 205. Painlevé, 206, 207. Peirce, 183, 186. Taylor, 40. Picard, 193, 194, 206. Thomson and Tait, 183, 185, 186. Pockels, 207. Todhunter, 106, 183. Poincaré, 207. Poisson, 186. Weierstrass, 193, 203. Price, 206. Williamson, 4, 52, 149, 183, 184. INDEX OF SUBJECTS. (The numbers refer to pages.) Applications to geometry, 50-58, 121, Equation, Equations. 134, 140, 141, 158. Invariants of, 204. mechanics, 58, 122, 144. Laplace's, 182. physics, 62, 122, 126, 144, 145. Legelndre's, 90, 105. Auxiliary equation, 65-67, 175. Linear, ordinary, 26, 28, 63, 64, 70, 82-84, 90, 101, 128. Bessel's equation, 105, 106. Linear, partial, 153-158, 173. Linear, simultaneous, 128-133. Characteristic, 151. Monge's, 171. Charpit, method of, 166. Non-homogelneous, 16, 179. Clairaut's equation, 36, 44. not decomposable, 32. Complementary function, 63, 174, 179. of cuspidal locus, 47-49. Condition that equation be exact, 18, of envelope, 42. 92, 197. of hypergeometric series, 105, 107. Constants of integration, 2, 149, 194, of nodal locus, 45, 48, 49. 195. of second order, 109-119. Criterion for independence of, con- of tac-locus, 45, 48, 49. stants, 195. Partial. integrals, 197. Partial, definition, derivation, 2, of integrability, 136, 138, 200. 146, 148. Cuspidal locus, 47. Partial, linear, 153-158, 169-187. Partial, linear homogeneous, 174 -Derivation, of ordinary equations, 4. 178. of partial equations, 146, 148. Partial, linear non-homogeneous, Discriminant, 40. 179-181. Partial, non-linear, integrals of, 149 -Envelope, 41-44, 150. 153. Equation, Equations. Partial, of first order, 149-169. Auxiliary, 65-67, 175. Partial, of second and higher orders, Bessel's, 105, 106. 169-187. Clairaut's, 36, 44. Poisson's, 186. Decomposable, 31. Reduction to equivalent system, 189. Definitions, 1, 17, 92, 146. Riccati's, 105, 106. Derivation of, 4, 146, 148. Simultaneous, 128-134. Exact, 17-19, 92-94, 197. Single integrable, 136, 138, 200. Geometrical meaning, 8-10, 134, Single non-integrable, 142. 140, 142, 158. Transformation of, 28, 90, 114, 115, Homogeneous, 15, 35, 82-84, 90,138, 117, 182. 174. Works on, 205-207. 233 234 INDEX OF SUBJECTS. Existence Theorem, 190. Monge's equations, method, 171. Factors, integrating, 21-26. Nodal locus, 45, 48, 49. Geometrical meaning, 8-10, 134, 140, Particular integrals. See integral. 142, 158. Physics, applications to. See applicaGeometry. See applications. tions. Poisson's equation, 186. Homogeneous. See equation. Hypergeometric series, equation of, Reduction of equations to equivalent 105, 107. system, 189. Relation between integrals, 111. Integrability, criterion of, 136, 138, Relation between integrals and coeffi200. cients, 199. Integral, integrals; also see solution. Removal of second term, 115. and coefficients, 199. Riccati's equation, 105, 106. complete, 64, 149. first, 94. Series, equation of hypergeometric, general, 150. 105, 107. of linear partial equations, 153. integration in, 101. of simultaneous equations, 133. Solutions, 2, 6; also see integrals. particular, 6, 64, 73-80, 87-90, 149, Spherical harmonics, 183, 184. 176, 180. Standard forms, 159-165. relation between, 111. Summary, 38, 48. singular, 150. Symbol D, 67, 68. Integration, constants of. See con- Symbolic fiction 1 70. starts. f(D) Integrating factors, 21-26. Symbolic functions f (), 1 86. Invariants, 204. f (6) Symbolic functions f(D, D'), Lagrange's solution, 154, 155. fym(D, D,)' Lagrangean lines, 155. 174, 176. Laplace's equation, 182. Legendre's equation, 105. Tac-locus, 45, 48, 49. Locus, 8, 42, 45, 47-49, 141, 151. Theories, modern, 202. Trajectories, 55-57. Mechanics, applications to. See appli- Transformations. See equations. cations. Modern theories, 202. Works on differential equations, 205. LONGMANS, GREEN, & CO.'S PUBLICA TIONS. THE HARPUR EUCLID. An Edition of Euclid's Elements revised in accordance with the Reports of the Cambridge Board of Mathematical Studies and the Oxford Board of the Faculty of Natural Science. 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