Book:George F. Simmons/Differential Equations
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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