Preface to the Second Edition
Preface to the First Edition
Suggestions for the Instructor
1 The Nature of Differential Equations. Separable Equations
1. Introduction
2. Gemeral Remarks on Solutions
3. Families of Curves. Orthogonal Trajectories
4. Growth, Decay, Chemical Reactions, and Mixing
5. Falling Bodies and Other Motion Problems
6. The Brachistochrone. Fermat and the Bernoullis
2 First Order Equations
7. Homogeneous Equations
8. Exact Equations
9. Integrating Factors
10. Linear Equations
11. Reduction of Order
12. The Hanging Chain. Pursuit Curves
13. Simple Electric Circuits
3 Second Order Linear Equations
14. Introduction
15. The General Solution of the Homogeneous Equation
16. The Use of a Known Solution to Find Another
17. The Homogeneous Equation with Constant Coefficients
18. The Method of Undetermined Coefficients
19. The Method of Variation and Parameters
20. Vibrations in Mechanical and Electrical Systems
21. Newton's Law of Gravitation and the Motions of the Planets
22. Higher Order Linear Equations. Coupled Harmonic Oscillators
23. Operator Methods for Finding Particular Solutions
Appendix A. Euler
Appendix B. Newton
4 Qualitative Properties of Solutions
24. Oscillations and the Sturm Separation Theorem
25. The Sturm Comparison Theorem
5 Power Series Solutions and Special Functions
26. Introduction. A Review of Power Series
27. Series Solutions of First Order Equations
28. Second Order Linear Equations. Ordinary Points
29. Regular Singular Points
30. Regular Singular Points (Continued)
31. Gauss's Hypergeometric Equation
32. The Point at Infinity
Appendix A. Two Convergence Proofs
Appendix B. Hermite Polynomials and Quantum Mechanics
Appendix C. Gauss
Appendix D. Chebyshev Polynomials and the Minimax Property
Appendix E. Riemann's Equation
6 Fourier Series and Orthogonal Functions
33. The Fourier Coefficients
34. The Problem of Convergence
35. Even and Odd Functions. Cosine and Sine Series
36. Extension to Arbitrary Intervals
37. Orthogonal Functions
38. The Mean Convergence of Fourier Series
Appendix A. A Pointwise Convergence Theorem
7 Partial Differential Equations and Boundary Value Problems
39. Introduction. Historical Remarks
40. Eigenvalues, Eigenfunctions, and the Vibrating String
41. The Heat Equation
42. The Dirichlet Problem for a Circle. Poisson's Integral
43. SturmLiouville Problems
Appendix A. The Existence of Eigenvalues and Eigenfunctions
8 Some Special Functions of Mathematical Physics
44. Legendre Polynomials
45. Properties of Legendre Polynomials
46. Bessel Functions. The Gamma Function
47. Properties of Bessel functions
Appendix A. Legendre Polynomials and Potential Theory
Appendix B. Bessel Functions and the Vibrating Membrane
Appendix C. Additional Properties of Bessel Functions
9 Laplace Transforms
48. Introduction
49. A Few Remarks on the Theory
50. Applications to Differential Equations
51. Derivatives and Integrals of Laplace Transforms
52. Convolutions and Abel's Mechanical Problem
53. More about Convolutions. The Unit Step and Impulse Functions
Appendix A. Laplace
Appendix B. Abel
10 Systems of First Order Equations
54. General Remarks on Systems
55. Linear Systems
56. Homogeneous Linear Systems with Constant Coefficients
57. Nonlinear Systems. Volterra's PreyPredator Equations
11 Nonlinear Equations
58. Autonomous Systems. The Phase Plane and Its Phenomena
59. Types of Critical Points. Stability.
60. Critical Points and Stability for Linear Systems
61. Stability by Liapunov's Direct Method
62. Simple Critical Points of Nonlinear Systems
63. Nonlinear Mechanics. Conservative Systems
64. Periodic Solutions. The PoincaréBendixson Theorem
Appendix A. Poincaré
Appendix B. Proof of Liénard's Theorem
12 The Calculus of Variations
65. Introduction. Some Typical Problems of the Subject
66. Euler's Differential Equation for an Extremal
67. Isoperimetric problems
Appendix A. Lagrange
Appendix B. Hamilton's Principle and Its Implications
13 The Existence and Uniqueness of Solutions
68. The Method of Successive Approximations
69. Picard's Theorem
70. Systems. The Second Order Linear Equation
14 Numerical Methods
71. Introduction
72. The Method of Euler
73. Errors
74. An Improvement to Euler
75. HigherOrder Methods
76. Systems
Numerical Tables
Answers
Index

