# Book:George F. Simmons/Differential Equations

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## George F. Simmons:

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

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

- ISBN 0-07-099572-9

### Subject Matter

### 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

*Answers**Index*

## Further Editions

## Source work progress

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $\S 3$: Appendix $\text B$: Newton

Second pass through:

- 1972: George F. Simmons:
*Differential Equations*... (previous) ... (next): $1$: The Nature of Differential Equations: Miscellaneous Problems for Chapter $1$: Problem $15$

Still not done:

- $\S 1.2$: Problems
- $\S 1.3$: Problems $3$ and $4$
- $\S 1.4$: Some of the historical examples, Problems $2$ to $4$
- $\S 1.5$: ProblemS $2$, $4$
- $\S 1$: most of the miscellaneous problems