A Course of Mathematical Analysis, Volume II, Part II

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AUTHOR'S PREFACE SECOND FRENCH EDITION The first part of this volume has undergone only slight changes, while the rather important modifications that have been made appear only in the last chapters. In the first edition I was able to devote but a few pages to par- tial differential equations of the second order and to the calculus of variations. In order to present in a less summary manner such broad subjects, I have concluded to defer them to a third volume, which will contain also a sketch of the recent theory of integral equations. The suppression of the last chapter has enabled me to make some additions, of which the most important relate to linear differential equations and to partial differential equations of the firSt Order " E. GOUESAT iii TRANSLATORS' PREFACE As the title indicates, the present volume is a translation of the first half of the second volume of Goursat's "Cours d' Analyse." The decision to publish the translation in two parts is due to the evi- dent adaptation of these two portions to the introductory courses in American colleges and universities in the theory of functions and in differential equations, respectively. After the cordial reception given to the translation of Goursat's first volume, the continuation was assured. That it has been delayed so long was due, in the first instance, to our desire to await the appearance of the second edition of the second volume in French. The advantage in doing so will be obvious to those who have observed the radical changes made in the second (French) edition of the second volume. Volume I was not altered so radi- cally, so that the present English translation of that volume may be used conveniently as a companion to this ; but references are given here to both editions pf the first volume, to avoid any possible difficulty in this connection. Our thanks are due to Professor Goursat, who has kindly given us his permission to make this translation, and has approved of the plan of publication in two parts. He has also seen all proofs in English and has approved a few minor alterations made in transla- tion as well as the translators' notes. The responsibility for the latter rests, however, with the translators. E. R. HEDRICK OTTO DUNKEL CONTENTS PAGE CHAPTER I. ELEMENTARY METHODS OF INTEGRATION . . 3 I. Formation of Differential Equations 3 1. Elimination of constants 3 II. Equations of the First Order 6 2. Separation of the variables 6 3. Homogeneous equations 8 4. Linear equations 9 5. Bernoulli's equation 11 6. Jacobi's equation 11 7. Riccati's equation 12 8. Equations not solved for y' 14 9. Lagrange's equation 16 10. Clairaut's equation 17 11. Integration of the equations F(x, y') = 0, F(y, y') = . . 18 12. Integrating factors 19 13. Application to conformal representation 22 14. Euler's equation 23 15. A method deduced from Abel's theorem 28 16. Darboux's theorems 29 17. Applications 32 III. Equations of Higher Order 35 18. Integration of the equation d n y/dx n =f(x) 35 19. Various* cases of depression 36 20. Applications 39 Exercises 42 CHAPTER II. EXISTENCE THEOREMS ........ 45 I. Calculus of Limits 45 21. Introduction 45 22. Existence of the integrals of a system of differential equations 45 23. Systems of linear equations 50 24. Total differential equations 51 25. Application of the method of the calculus of limits to partial differential equations 53 26. The general integral of a system of differential equations . 57 IL The Method of Successive Approximations. The Cauchy-Lipschitz Method 61 27. Successive approximations 61 28. The case of linear equations 64 29. Extension to analytic functions 66 30. The Cauchy-Lipschitz method . . 68 III. First Integrals. Multipliers 74 31. First integrals 74 32. Multipliers 81 33. Invariant integrals • • 83 IV. Infinitesimal Transformations 86 34. One-parameter groups 86 35. Application to differential equations ........ 89 36. Infinitesimal transformations 91 Exercises 98 CHAPTEE III. LINEAR DIFFERENTIAL EQUATIONS . . . 100 I. General Properties. Fundamental Systems . . . 100 37. Singular points of a linear differential equation .... 100 38. Fundamental systems - 102 39. The general linear equation 106 40. Depression of the order of a linear equation ...... 109 41. Analogies with algebraic equations,. . . 113 42. The adjoint equation 115 II. The Study of Some Particular Equations . , . 117 43. Equations with constant coefficients . .• V 117 44. D'Alembert's method . 122 45. Euler's linear equation 123 46. Laplace's equation . • III. Regular Integrals. Equations with Periodic Coefficients 128 47. Permutation of the integrals around a critical point . . • '. 129 48. Examination of the general case . . . . . ... 131 49. Formal expressions for the integrals . I . . . . . ••■ 133 CONTENTS vii PAGE 50. Fuchs' theorem . . 134 51. Gauss's equation 140 52. Bessel's equation 142 53. Picard's equations . . . . ... . . 143 54. Equations with periodic coefficients ........ 146 55. Characteristic exponents 150 IV. Systems of Linear Equations 152 56. General properties 152 57. Adjoint systems 156 58. Linear systems with constant coefficients ....... 157 59. Reduction to a canonical form . . 161 60. Jacobi's equation 163 61. Systems with periodic coefficients . 164 62. Reducible systems . . . . 165 Exercises 167 CHAPTER IV. NON-LINEAR DIFFERENTIAL EQUATIONS . . 172 I. Exceptional Initial Values . . . . . . . . . . 172 63. The case where the derivative becomes infinite . . . / 172 64. Case where the derivative is indeterminate 173 II. A Study of Some Equations of the Eirst Order 180 66. Singular points of integrals ....... , ... 180 67. Functions defined by a differential equation i/' = R (x, y) . 182 68. Single-valued functions deduced from the equation 0O m = R(y) 187 69. Existence of elliptic functions deduced from Euler's equation 194 70. Equations of higher order 196 III. Singular Integrals 198 71. Singular integrals of ah equation of the first order . . . 198 72. General comments 204 73. Geometric interpretation 207 74. Singular integrals of systems of differential equations . . 208 Exercises ■ . . 212 CHAPTER V. PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 214 I. Linear Equations of the Eirst Order 214 75. General method 214 76. Geometric interpretation 218 77. Congruences of characteristic curves 222 viii CONTENTS PAGE II. Total Differential Equations 225 78. The equation dz = Adx + Bdy . * 225 79. Mayer's method .229 80. The equation Pdx + Qdy + Rdz = 230 81. The parenthesis (u, v) and the bracket [w, v~\ ..... 234 III. Equations of the Eirst Order in Three Variables 236 82. Complete integrals 236 83. Lagrange and Charpit's method 240 84. Cauchy's problem 246 85. Characteristic curves. Cauchy's method 249 86. The characteristic curves derived from a complete integral 259 87. Extension of Cauchy's method 261 IV. Simultaneous Equations 265 88. Linear homogeneous systems 265 89. Complete systems 267 90. Generalization of the theory of the complete integrals . . 272 91. Involutory systems 274 92. Jacobi's method 277 V. Generalities on the Equations of Higher Order 278 93. Elimination of arbitrary functions 278 94. General existence theorem 283 Exercises 287 Index 291