Chapter 1．Definitions and Elementary Problems 1
1．General remarks 1
2．Differential equation．Order．Degree 2
3．Solution of a differential equation 3
4．Geometric considerations 5
5．Existence theorems 7
6．General solution．Particular solution 8
7．Finding differential equation from general solution 8
8．Variables separable 11
9．Review exercises 13
Chapter 2．Applications 15
10．Geometric applications using rectangular coordinates 15
11．Orthogonal trajectories 16
12．Geometric applications using polar coordinates 18
13．Use of limits 20
14．Physical applications 21
15．Compound-interest-law problems 22
16．Acceleration．Velocity．Distance 24
17．Other rate problems 27
18．Miscellaneous problems 30
Chapter 3．Differential Equations of the First Order and the First Degree 33
19．Simple substitutions 33
20．Homogeneous equations 35
21．Equations of the form(ax+by+c)dx+(αx+βy+γ)dy=0 37
22．Exact differentials 38
23．Exact differential equations 39
24．Integrating factors 42
25．Linear differential equation 45
26．Equations reducible to linear form 48
27．Simultaneous equations 49
28．Summary 51
Chapter 4．Applications Involving Differential Equations of the First Order 54
29．Miscellaneous elementary applications 54
30．Applications involving simultaneous equations 56
31．Applications to the flow of electricity 59
32．Air pressure 60
33．Applications involving forces and velocities 61
34．Review problems 67
Chapter 5．First-order Equations of Degree Higher than the First 70
35．Foreword 70
36．Equations solvable for dy/dx 70
37．Envelopes 72
38．Envelope from differential equation 75
39．Equations solvable for y 78
40．Equations solvable for x 80
41．Review problems 81
Chapter 6．Linear Differential Equations with Constant Coefficients 83
42．Operators 83
43．Linear independence of functions 86
44．Linear differential equation 88
45．Homogeneous linear differential equation with constant coefficients 89
46．Auxiliary equation has repeated roots 90
47．Constants of integration from initial conditions 91
48．Auxiliary equation has imaginary roots 92
49．Right-hand member not zero 94
50．Special case when the right-hand member is not zero 96
51．A basic theorem relating to operators 97
52．Methods using symbolic operators 98
53．Variation of parameters 103
54．Simultaneous differential equations 106
55．Summary and review exercises 108
Chapter 7．Laplace Transforms 110
56．Introduction 110
57．Definition of a Laplace transform 110
58．Some properties of Laplace transforms 111
59．Deriving transform relations from given ones 114
60．Inverse transforms of products 117
61．Transforms of derivatives 121
62．Solving differential equations by transforms 122
63．Solving systems of differential equations 124
64．Resolving a fraction into partial fractions 126
65．Fractions having repeated factors in the denominator 128
66．Partial fractions．Quadratic factors 130
67．Review problems 133
Chapter 8．Applications of Linear Equations with Constant Coefficients 135
68．Harmonic motion．Damping 135
69．Types of damping．Resonance 137
70．Forces．Accelerations．Moments 139
71．Some fundamental equations of motion 140
72．Oscillatory motion 141
73．Plane motions of bodies 147
74．Kirchhoff＇s current law and electromotive-force law 151
75．Simple circuits containing constant electromotive force 153
76．Simple circuits containing a sinusoidal electromotive force 154
77．Resonance 155
78．Applications of Kirchhoff＇s laws to networks 156
79．Review problems 161
Chapter 9．Miscellaneous Differential Equations of Order Higher than the First 163
80．Reduction of order by substitution 163
81．Dependent variable absent 163
82．Independent variable absent 165
83．Method based on factorization of the operator 166
84．Euler＇s linear equation 169
85．Second-order linear equation 170
86．Review exercises 171
Chapter 10．Applications 173
87．Radius of curvature 173
88．Cables．The catenary 174
89．Equation of elastic curve．Beams 176
90．Columns 181
91．Motion of a particle in a plane 182
92．Review problems 185
Chapter 11．Existence Theorems and Applications 187
93．Foreword 187
94．Replacement of differential equations by a system of the first order and first degree 187
95．Existence theorems 188
96．Differential equations of the first order and first degree in the unknowns 191
97．Total differential equations 194
98．Geometrical interpretation 198
99．Fields of force in space 198
100．Review exercises 200
Chapter 12．Solution by Series 202
101．Introduction 202
102．Integration in series 204
103．Solution involving a more general type of series 208
104．Indicial equation has roots differing by an integer 210
105．The gamma function 213
106．Bessel＇s equation 215
107．Bessel＇s functions 217
108．Expansion of functions in terms of Bessel＇s functions 222
109．Legendre＇s functions 224
Chapter 13．Numerical Solutions of Differential Equations 229
110．Introduction 229
111．Methods of successive approximations 229
112．Newton＇s interpolation formula 231
113．Interpolation 234
114．Formulas for approximate integration 236
115．Illustration and discussion of formulas(16)to(21)，§114 237
116．Halving the interval h of x 240
117．Numerical solution of a system of simultaneous equations 242
118．The Runge-Kutta method 245
Chapter 14．Partial Differential Equations 248
119．Introduction 248
120．Solution of a partial differential equation 248
121．Equations easily integrable 250
122．Equations having the form Pp+Qq=R 251
123．Finding particular solutions satisfying given conditions 254
124．Separation of variables 256
125．Hyperbolic，parabolic，elliptic equations 258
Chapter 15．Applications of Partial Differential Equations 263
126．Fourier series 263
127．Cosine series．Sine series 267
128．Application to nuclear fission 269
129．Vibrations of a string 272
130．Vibrations of a rod 275
131．Flow of heat 276
132．One-dimensional heat flow 279
133．Vibrations of a membrane 281
134．Telephone，telegraph，and radio equations 283
135．Fluid motion 286
Answers 291
Index 313