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
Author(s): Édouard Goursat, Earle Raymond Hedrick and Otto Dunkel
Edition: 1
Publisher: Ginn & company
Year: 1917
Language: English
Pages: 306
City: Boston, New York
Tags: Calculus, Differential Equations
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
