Differential Equations Book

I The existence and uniqueness theorem

1 Résumé of some elementary theory of differential equations 1

2 Preliminaries to the fundamental theorem 4

3 The existence and uniqueness theorem for normal differential systems 5

4 Additional remarks 10

5 Circular functions 13

6 Elliptic functions 19

II The behaviour of the characteristics of a first-order equation

7 Preliminary considerations 27

8 Examples of equations with singular points 32

9 Study of the abridged equation 39

10 Some theorems of a general character 45

11 The Poincaré index 53

12 The node 55

13 The focus and the col 63

14 Limit cycles and relaxation oscillations 74

15 Periodic solutions in the phase space 82

III Boundary problems for linear equations of the second order

16 Preliminary considerations 89

17 A theorem of de la Vallée Poussin 92

18 Simplifications of the given equation 96

19 Theorems oh the zeros and on the maxima and minima of integrals 98

20 Comparison theorems and their corollaries 101

21 The interval between successive zeros of an integral 104

22 An important change of variable 107

23 The oscillation theorem 112

24 Eigenvalues and eigenfunctions 117

25 A physical interpretation 119

26 Some properties of eigenvalues and eigenfunctions 123

27 Connection with the theory of integral equations 132

IV Asymptotic methods

28 General remarks 139

29 A general method applicable to linear differential equations 142

30 Differential equations with stable integrals 148

31 The case in which the coefficient of y tends to a negative limit 154

32 Preliminaries to the asymptotic treatment of eigenvalues and of eigenfunctions 163

33 First form of asymptotic expression for the eigenfunctions 166

34 Asymptotic expression for the eigenvalues 169

35 Second form of asymptotic expression for the eigenfunctions 174

36 Equations with transition points 177

37 The Laguerre differential equation and polynomials 180

38 Asymptotic behaviour of the Laguerre polynomials 186

39 The Legendre differential equation and polynomials 191

40 An asymptotic expression for the Legendre polynomials 195

V Differential equations in the complex field

41 Majorizing functions 202

42 Proof of the fundamental theorem by Cauchy's method 205

43 General remarks on singular points of solutions of differential equations. The case of linear equations 210

44 Investigation of the many-valuedness of integrals of a linear equation 214

45 The case with no essential singularities 218

46 Integration in series of equations of Fuchs' type 221

47 Totally Fuchsian equations. The hypergeometric equation 228

48 Preliminary remarks on points of essential singularity 240

49 An application of the method of successive approximations 245

50 'Asymptotic integration' of the reduced equation 249

51 Conclusion and further comments 253

52 Application to confluent hypergeometric functions and to Bessel functions 257

Bibliography 265

Author index 269

General index 271