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Differential Equations with Applications and Historical Notes

George F Simmons
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

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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. Sturm-Liouville 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 Prey-Predator 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 Poincaré-Bendixson 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. Higher-Order Methods

76. Systems

Numerical Tables