Cover......Page 1
Half Title......Page 2
Title Page......Page 4
Copyright Page......Page 5
Dedication......Page 6
Table of Contents......Page 8
Preface to the Third Edition......Page 12
Preface to the Second Edition......Page 14
Preface to the First Edition......Page 16
Suggestions for the Instructor......Page 20
About the Author......Page 22
1 Introduction......Page 24
2 General Remarks on Solutions......Page 27
3 Families of Curves. Orthogonal Trajectories......Page 34
4 Growth, Decay, Chemical Reactions, and Mixing......Page 42
5 Falling Bodies and Other Motion Problems......Page 54
6 The Brachistochrone. Fermat and the Bernoullis......Page 63
Appendix A: Some Ideas From the Theory of Probability: The Normal Distribution Curve (or Bell Curve) and Its Differential Equation......Page 74
7 Homogeneous Equations......Page 88
8 Exact Equations......Page 92
9 Integrating Factors......Page 97
10 Linear Equations......Page 104
11 Reduction of Order......Page 108
12 The Hanging Chain. Pursuit Curves......Page 111
13 Simple Electric Circuits......Page 118
14 Introduction......Page 130
15 The General Solution of the Homogeneous Equation......Page 136
16 The Use of a Known Solution to find Another......Page 142
17 The Homogeneous Equation with Constant Coefficients......Page 145
18 The Method of Undetermined Coefficients......Page 150
19 The Method of Variation of Parameters......Page 156
20 Vibrations in Mechanical and Electrical Systems......Page 159
21 Newton’s Law of Gravitation and The Motion of the Planets......Page 169
22 Higher Order Linear Equations. Coupled Harmonic Oscillators......Page 178
23 Operator Methods for Finding Particular Solutions......Page 184
Appendix A. Euler......Page 193
Appendix B. Newton......Page 202
24 Oscillations and the Sturm Separation Theorem......Page 210
25 The Sturm Comparison Theorem......Page 217
26 Introduction. A Review of Power Series......Page 220
27 Series Solutions of First Order Equations......Page 229
28 Second Order Linear Equations. Ordinary Points......Page 233
29 Regular Singular Points......Page 242
30 Regular Singular Points (Continued)......Page 252
31 Gauss’s Hypergeometric Equation......Page 259
32 The Point at Infinity......Page 265
Appendix A. Two Convergence Proofs......Page 269
Appendix B. Hermite Polynomials and Quantum Mechanics......Page 273
Appendix C. Gauss......Page 285
Appendix D. Chebyshev Polynomials and the Minimax Property......Page 293
Appendix E. Riemann’s Equation......Page 301
33 The Fourier Coefficients......Page 312
34 The Problem of Convergence......Page 324
35 Even and Odd Functions. Cosine and Sine Series......Page 333
36 Extension to Arbitrary Intervals......Page 342
37 Orthogonal Functions......Page 348
38 The Mean Convergence of Fourier Series......Page 359
Appendix A. A Pointwise Convergence Theorem......Page 368
39 Introduction. Historical Remarks......Page 374
40 Eigenvalues, Eigenfunctions, and the Vibrating String......Page 378
41 The Heat Equation......Page 389
42 The Dirichlet Problem for a Circle. Poisson’s Integral......Page 395
43 Sturm–Liouville Problems......Page 402
Appendix A. The Existence of Eigenvalues and Eigenfunctions......Page 411
44 Legendre Polynomials......Page 416
45 Properties of Legendre Polynomials......Page 423
46 Bessel Functions. The Gamma Function......Page 430
47 Properties of Bessel Functions......Page 441
Appendix A. Legendre Polynomials and Potential Theory......Page 450
Appendix B. Bessel Functions and the Vibrating Membrane......Page 458
Appendix C. Additional Properties of Bessel Functions......Page 464
48 Introduction......Page 470
49 A Few Remarks on the Theory......Page 475
50 Applications to Differential Equations......Page 480
51 Derivatives and Integrals of Laplace Transforms......Page 486
52 Convolutions and Abel’s Mechanical Problem......Page 491
53 More about Convolutions. The Unit Step and Impulse Functions......Page 498
Appendix A. Laplace......Page 506
Appendix B. Abel......Page 507
54 General Remarks on Systems......Page 510
55 Linear Systems......Page 514
56 Homogeneous Linear Systems with Constant Coefficients......Page 521
57 Nonlinear Systems. Volterra’s Prey-Predator Equations......Page 530
58 Autonomous Systems. The Phase Plane and Its Phenomena......Page 536
59 Types of Critical Points. Stability......Page 542
60 Critical Points and Stability for Linear Systems......Page 552
61 Stability By Liapunov’s Direct Method......Page 564
62 Simple Critical Points of Nonlinear Systems......Page 570
63 Nonlinear Mechanics. Conservative Systems......Page 580
64 Periodic Solutions. The Poincaré–Bendixson Theorem......Page 586
65 More about the van der Pol Equation......Page 595
Appendix A. Poincaré......Page 597
Appendix B. Proof of Liénard’s Theorem......Page 599
66 Introduction. Some Typical Problems of the Subject......Page 604
67 Euler’s Differential Equation for an Extremal......Page 607
68 Isoperimetric Problems......Page 618
Appendix A. Lagrange......Page 629
Appendix B. Hamilton’s Principle and Its Implications......Page 631
69 The Method of Successive Approximations......Page 644
70 Picard’s Theorem......Page 649
71 Systems. The Second Order Linear Equation......Page 661
72 Introduction......Page 666
73 The Method of Euler......Page 669
74 Errors......Page 673
75 An Improvement to Euler......Page 675
76 Higher Order Methods......Page 680
77 Systems......Page 684
Numerical Tables......Page 690
Answers......Page 704
Index......Page 746