Differential Equations: An Introduction with Applications

Title Page......Page 3
Copyright Page......Page 4
Contents......Page 5
Preface (to the 5th German edition)......Page 11
Preface to the 6th edition......Page 13
Preface to the English edition......Page 15
1 Definitions and notation......Page 17
2 Examples of differential equations arising in physics......Page 18
3 Solution curves in the direction field......Page 21
4 Separation of the variables......Page 23
5 The similarity differential equation......Page 26
6 Simple cases reducible to the similarity differential equation......Page 27
7 Homogeneous and inhomogeneous equations; the trivial solution......Page 29
9 Solution of the inhomogeneous equation......Page 30
10 Reduction to a linear differential equation......Page 33
11 The Riccati differential equation......Page 34
12 Exact differential equations......Page 35
13 Integrating factors......Page 36
14 Single-valued and multi-valued direction fields......Page 38
15 Non-uniqueness of the solution......Page 39
16 The Lipschitz condition; its stronger and weaker forms......Page 40
17 The method of successive approximations......Page 42
18 The existence theorem......Page 45
19 Proof of uniqueness......Page 48
20 Systems of differential equations and a differential equation of  nth order......Page 49
21 Some fundamental concepts in functional analysis......Page 52
22 Banach\'s fixed-point theorem and the existence theorem for ordinary differential equations......Page 58
23 Regular and singular line-elements. Definitions and examples......Page 61
24 Isolated singular points......Page 65
25 On the theory of isolated singular points......Page 68
26 Clairaut\'s and d\'Alembert\'s differential equations......Page 71
27 Oscillations with one degree of freedom. Phase curves......Page 74
28 Examples of oscillations and phase curves......Page 76
29 Periodic oscillations of an autonomous undamped system with  one degree of freedom......Page 82
30 Problems......Page 87
31 Solutions......Page 89
1 The dependent variable y does not appear explicitly......Page 93
2 The equation y\" = f(y) and the energy integral......Page 94
3 The general differential equation in which the independent variable x does not occur explicitly......Page 95
4 The differential equation contains only the ratios y(v)/y......Page 97
5 Notation......Page 98
6 The superposition theorem......Page 99
7 Reduction of the order of a linear differential equation......Page 100
8 Linear dependence of functions......Page 101
9 The Wronskian criterion for linear independence of functions......Page 104
10 The general solution of a linear differential equation......Page 106
11 A trial substitution for the solution; the characteristic equation......Page 108
12 Multiple zeros of the characterisitc equation......Page 111
14 The equation for forced oscillations......Page 115
15 Solution of the homogeneous equation for oscillations......Page 117
4 Determination of a particular solution of the inhomogeneous linear differential equation......Page 119
16 The method of variation of the constants......Page 120
17 The rule-of-thumb method......Page 121
18 Introduction of a complex differential equation......Page 123
20 The substitution for the solution; the characteristic equation......Page 125
21 Examples......Page 127
22 Example: vibrations of a motorcar; (types of coupling)......Page 128
23 The fundamental system of solutions......Page 132
25 Matrix A constant; characteristic roots of a matrix......Page 134
26 The three main classes in the theory of square matrices......Page 135
27 Application to the theory of oscillations......Page 137
28 Example of a physical system which leads to a non-normalizable matrix......Page 138
29 Transformation of a normal or normalizable matrix to diagonal form......Page 140
30 Engineering examples leading to differential equations with periodic coefficients......Page 146
31 Periodic solutions of the homogeneous system......Page 148
32 Stability......Page 149
33 Periodic solutions for the inhomogeneous system......Page 151
34 An example for the stability theory......Page 152
35 Definition of the Laplace transformation......Page 154
36 Differentiation and integration of the original element......Page 157
37 Using the Laplace transformation to solve initial-value problems for ordinary differential equations......Page 159
38 Transients and periodic solutions......Page 162
39 The convolution theorem and integral equations of the first kind......Page 168
40 The inverse Laplace transformation and tables......Page 172
1 Initial-value problems and boundary-value problems......Page 175
2 A beam. The several fields of the differential equation......Page 176
3 The number of solutions in linear boundary-value problems......Page 179
4 The differential equation of a catenary......Page 180
5 The differential equation y\" = y2......Page 183
6 A countable infinity of solutions of a boundary-value problem with the equation y\" = - y3......Page 187
7 Semi-homogeneous and fully homogeneous boundary-value problems......Page 189
8 The general alternative......Page 190
9 Some very simple examples of Green\'s functions......Page 192
10 The Green\'s function as an influence function......Page 194
11 General definition of the Green\'s function......Page 196
12 The solution formula for the boundary-value problem......Page 199
13 The completely homogeneous boundary-value problem......Page 200
14 The corresponding non-linear boundary-value problem......Page 204
15 Partial differential equations......Page 205
16 The Bernoulli substitution for natural vibrations......Page 207
17 Self-adjoint and positive-definite eigenvalue problems......Page 210
18 Orthogonality of the eigenfunctions......Page 213
19 Orthonormal systems......Page 215
20 Approximation in the mean......Page 220
21 On the expansion theorem......Page 221
22 The theorem on inclusion of an eigenvalue between two quotients......Page 223
23 Some simple examples......Page 226
24 The Euler equation in the calculus of variations in the simplest case......Page 229
25 Free boundary-value problems and the calculus of variations......Page 232
27 Non-linear eigenvalue problems and branching problems......Page 234
28 Example. The branching diagram for a Urysohn integral equation......Page 235
29 Problems......Page 237
30 Solutions......Page 240
1 Solutions of Laplace\'s equation......Page 247
2 The generating function......Page 249
3 Legendre functions of the second kind......Page 253
4 Another explicit representation of the Legendre polynomials......Page 254
5 Orthogonality......Page 255
6 The partial differential equation for vibrations of a membrane......Page 256
7 The Bernoulli substitution for vibrations of a membrane......Page 257
8 The generating function......Page 259
9 Deductions from the generating function......Page 260
10 An integral representation......Page 262
11 An example from astronomy: the Kepler equation......Page 263
13 More general differential equations giving Bessel functions......Page 265
3 Series expansions; the hypergeometric function......Page 270
15 The series substitution; the indicial equation......Page 271
16 The roots of the indicial equation......Page 272
17 Examples; the hypergeometric function......Page 273
18 The method of perturbations and singular points......Page 276
19 An example of the series-expansion technique......Page 278
1 Simple linear partial differential equations......Page 281
2 The wave equation and Laplace\'s equation......Page 283
3 Non-linear differential equations. The breaking of a wave......Page 286
4 The three fundamental types of a quasi-linear partial differential equation of the second order......Page 291
5 Range of influence and domain of continuation......Page 294
6 Solution of a boundary-value problem for a circular domain......Page 296
7 Example: a temperature distribution......Page 298
8 The boundary-maximum theorem in potential theory in the plane and in three-dimensional space......Page 302
9 Continuous dependence of the solution of the boundary-value problem on the data......Page 304
10 Monotonicity theorems. Optimization and approximation......Page 306
11 A numerical example; a torsion problem......Page 308
12 Well-posed and not well-posed problems......Page 310
13 Further examples of problems which are not well-posed......Page 312
14 Examples of free boundary-value problems......Page 313
15 The variational problem for the boundary-value problem of the first kind in potential theory......Page 315
16 The Ritz method......Page 317
17 The method of finite elements......Page 318
18 An example......Page 321
19 A parabolic equation (the equation of heat conduction)......Page 322
20 The Laplace transformation with the wave equation......Page 324
21 The reciprocity formulae for the Fourier transformation......Page 325
22 The Fourier transformation with the heat-conduction equation......Page 329
1 Preliminary remarks and some rough methods of approximation......Page 331
2 The Runge and Kutta method of approximate numerical integration......Page 332
3 The method of central differences......Page 334
4 The method of difference quotients......Page 337
5 Multi-point methods......Page 339
6 The Ritz method......Page 340
7 Miscellaneous problems on Chapter I......Page 342
8 Solutions......Page 344
9 Miscellaneous problems on Chapters II and III......Page 348
10 Solutions......Page 352
11 Miscellaneous problems on Chapters IV and V......Page 363
12 Solutions......Page 368
13 The dates of some mathematicians......Page 379
Bibliography......Page 380
Index......Page 383