Differential Equations with Applications and Historical Notes (2nd Edition)

Cover......Page __sk_0000.djvu
Copyright......Page __sk_0002.djvu
Contents......Page __sk_0007.djvu
Preface to the Second Edition......Page __sk_0011.djvu
Preface to the First Edition......Page __sk_0013.djvu
Suggestions for the Instructor......Page __sk_0017.djvu
1. Introduction......Page __sk_0023.djvu
2. General Remarks on Solutions......Page __sk_0026.djvu
3. Families of Curves. Orthogonal Trajectories......Page __sk_0032.djvu
4. Growth, Decay, Chemical Reactions, and Mixing......Page __sk_0039.djvu
5. Falling Bodies and Other Motion Problems......Page __sk_0051.djvu
6. The Brachistochrone. Fermat and the Bernoullis......Page __sk_0057.djvu
7. Homogeneous Equations......Page __sk_0069.djvu
8. Exact Equations......Page __sk_0073.djvu
9. Integrating Factors......Page __sk_0076.djvu
10. Linear Equations......Page __sk_0082.djvu
11. Reduction of Order......Page __sk_0085.djvu
12. The Hanging Chain. Pursuit Curves......Page __sk_0088.djvu
13. Simple Electric Circuits......Page __sk_0093.djvu
14. Introduction......Page __sk_0103.djvu
15. The General Solution of the Homogeneous Equation......Page __sk_0109.djvu
16. The Use of a Known Solution to Find Another......Page __sk_0114.djvu
17. The Homogeneous Equation with Constant Coefficients......Page __sk_0117.djvu
18. The Method of Undetermined Coefficients......Page __sk_0121.djvu
19. The Method of Variation of Parameters......Page __sk_0125.djvu
20. Vibrations in Mechanical and Electrical Systems......Page __sk_0128.djvu
21. Newton's Law of Gravitation and the Motion of the Planets......Page __sk_0137.djvu
22. Higher Order Linear Equations. Coupled Harmonic Oscillators......Page __sk_0144.djvu
23. Operator Methods for Finding Particular Solutions......Page __sk_0150.djvu
Appendix A. Euler......Page __sk_0158.djvu
Appendix B. Newton......Page __sk_0168.djvu
24. Oscillations and the Sturm Separation Theorem......Page __sk_0177.djvu
25. The Sturm Comparison Theorem......Page __sk_0183.djvu
26. Introduction. A Review of Power Series......Page __sk_0187.djvu
27. Series Solutions of First Order Equations......Page __sk_0194.djvu
28. Second Order Linear Equtions. Ordinary Points......Page __sk_0198.djvu
29. Regular Singular Points......Page __sk_0206.djvu
30. Regular Singular Points (Continued)......Page __sk_0214.djvu
31. Gauss's Hypergeometric Equation......Page __sk_0221.djvu
32. The Point at Infinity......Page __sk_0226.djvu
Appendix A. Two Convergence Proofs......Page __sk_0230.djvu
Appendix B. Hermite Polynomials and Quantum Mechanics......Page __sk_0233.djvu
Appendix C. Gauss......Page __sk_0243.djvu
Appendix D. Chebyshev Polynomials and the Minimax Property......Page __sk_0252.djvu
Appendix E. Riemann's Equation......Page __sk_0259.djvu
33. The Fourier Coefficients......Page __sk_0268.djvu
34. The Problem of Convergence......Page __sk_0279.djvu
35. Even and Odd Functions. Cosine and Sine Series......Page __sk_0287.djvu
36. Extension to Arbitrary Intervals......Page __sk_0294.djvu
37. Orthogonal Functions......Page __sk_0299.djvu
38. The Mean Convergence of Fourier Series......Page __sk_0307.djvu
Appendix A. A Pointwise Convergence Theorem......Page __sk_0315.djvu
39. Introduction. Historical Remarks......Page __sk_0320.djvu
40. Eigenvalues, Eigenfunctions, and the Vibrating String......Page __sk_0324.djvu
41. The Heat Equation......Page __sk_0333.djvu
42. The Dirichlet Problem for a Circle. Poisson's Integral......Page __sk_0339.djvu
43. Sturm-Liouville Problems......Page __sk_0345.djvu
Appendix A. The Existence of Eigenvalues and Eigenfunctions......Page __sk_0353.djvu
44. Legendre Polynomials......Page __sk_0357.djvu
45. Properties of Legendre Polynomials......Page __sk_0364.djvu
46. Bessel Functions. The Gamma Function......Page __sk_0370.djvu
47. Properties of Bessel Functions......Page __sk_0380.djvu
Appendix A. Legendre Polynomials and Potential Theory......Page __sk_0387.djvu
Appendix B. Bessel Functions and the Vibrating Membrane......Page __sk_0393.djvu
Appendix C. Additional Properties of Bessel Functions......Page __sk_0399.djvu
48. Introduction......Page __sk_0403.djvu
49. A Few Remarks on the Theory......Page __sk_0407.djvu
50. Applications to Differential Equations......Page __sk_0412.djvu
51. Derivatives and Integrals of Laplace Transforms......Page __sk_0416.djvu
52. Convolutions and Abel's Mechanical Problem......Page __sk_0421.djvu
53. More about Convolutions. The Unit Step and Impulse Functions......Page __sk_0427.djvu
Appendix A. Laplace......Page __sk_0434.djvu
Appendix B. Abel......Page __sk_0435.djvu
54. General Remarks on Systems......Page __sk_0439.djvu
55. Linear Systems......Page __sk_0443.djvu
56. Homogeneous Linear Systems with Constant Coefficients......Page __sk_0449.djvu
57. Nonlinear Systems. Volterra's Prey-Predator Equations......Page __sk_0456.djvu
58. Autonomous Systems. The Phase Plane and Its Phenomena......Page __sk_0462.djvu
59. Types of Critical Points. Stability......Page __sk_0468.djvu
60. Critical Points and Stability for Linear Systems......Page __sk_0477.djvu
61. Stability by Liapunov's Direct Method......Page __sk_0487.djvu
62. Simple Critical Points of Nonlinear Systems......Page __sk_0493.djvu
63. Nonlinear Mechanics. Conservative Systems......Page __sk_0502.djvu
64. Periodic Solutions. The Poincaré-Bendixson Theorem......Page __sk_0508.djvu
Appendix A. Poincaré......Page __sk_0516.djvu
Appcndix B. Proof of Liénard's Theorem......Page __sk_0519.djvu
65. Introduction. Some Typical Problems of the Subject......Page __sk_0524.djvu
66. Euler's Differential Equation for an Extremal......Page __sk_0527.djvu
67. Isoperimetric Problems......Page __sk_0537.djvu
Appendix A. Lagrange......Page __sk_0546.djvu
Appendix B. Hamilton's Principle and Its Implications......Page __sk_0548.djvu
68. The Method of Successive Approximations......Page __sk_0560.djvu
69. Picard's Theorem......Page __sk_0565.djvu
70. Systems. The Second Order Linear Equation......Page __sk_0574.djvu
71. Introduction......Page __sk_0578.djvu
72. The Method of Euler......Page __sk_0581.djvu
73. Errors......Page __sk_0585.djvu
74. An Improvement to Euler......Page __sk_0587.djvu
75. Higher-Order Methods......Page __sk_0591.djvu
76. Systems......Page __sk_0595.djvu
Numerical Tables......Page __sk_0599.djvu
Answers......Page __sk_0607.djvu
Index......Page __sk_0639.djvu