# Book:George F. Simmons/Differential Equations

## George F. Simmons: Differential Equations with Applications and Historical Notes

Published $\text {1972}$, McGraw-Hill

ISBN 0-07-099572-9.

### Contents

Preface
Suggestions for the Instructor
1 THE NATURE OF DIFFERENTIAL EQUATIONS
1. Introduction
2. General remarks on solutions
3. Families of curves. Orthogonal trajectories
4. Growth, decay, and chemical reactions
5. Falling bodies and other rate problems
6. The brachistchrone. 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
Appendix A. Numerical methods
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 of parameters
20. Vibrations in mechanical systems
21. Newton's law of gravitation and the motion of the planets
Appendix A. Euler
Appendix B. Newton
4 OSCILLATION THEORY AND BOUNDARY VALUE PROBLEMS
22. Qualitative properties of solutions
23. The Sturm comparison theorem
24. Eigenvalues, eigenfunctions, and the vibrating string
Appendix A. Regular Sturm-Liouville problems
5 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS
25. Introduction. A review of power series
26. Series solutions of first order equations
27. Second order linear equations. Ordinary points
28. Regular singular points
29. Regular singular points (continued)
30. Gauss's hypergeometric equation
31. 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 SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS
32. Legendre polynomials
33. Properties of Legendre polynomials
34. Bessel functions. The gamma function
35. Properties of Bessel functions
Appendix A. Legendre polynomials and potential theory
Appendix B. Bessel functions and the vibrating membrane
Appendix C. Additional propeqties of Bessel functions
7 SYSTEMS OF FIRST ORDER EQUATIONS
36. General remarks on systems
37. Linear systems
38. Homogeneous linear systems with constant coefficients
39. Nonlinear systems. Volterra's prey-predator equations
8 NONLINEAR EQUATIONS
40. Autonomous systems. The phase plane and its phenomena
41. Types of critical points. Stability
42. Critical points and stability for linear systems
43. Stability by Liapupov's direct method
44. Simple critical points of nonlinear systems
45. Nonlinear mechanics. Conservative systems
46. Periodic solutions. The Poincaré-Bendixson theorem
Appendix A. Poincaré
Appendix B. Proof of Liénard's theorem
9 THE CALCULUS OF VARIATIONS
47. Introduction. Some typical problems of the subject
48. Euler's differential equation for an extremal
49. Isoperimetric problems
Appendix A. Lagrange
Appendix B. Hamilton's principle and its implications
10 LAPLACE TRANSFORMS
50 Introduction
51. A few remarks on the theory
52. Applications to differential equations
53. Derivatives and integrals of Laplace transforms
54. Convolutions and Abel's mechanical problem
Appendix A. Laplace
Appendix B. Abel
11 THE EXISTENCE AND UNIQUENESS OF SOLUTIONS
55. The method of successive approximations
56. Picard's theorem
57. Systems. The second order linear equation