A revision of a much-admired text distinguished by the exceptional prose and historical/mathematical context that have made Simmons' books classics. The Second Edition includes expanded coverage of Laplace transforms and partial differential equations as well as a new chapter on numerical methods.

Table of Contents:
Preface to the Second Edition Preface to the First Edition Suggestions for the Instructor PART 1 THE NATURE OF DIFFERENTIAL EQUATIONS. SEPARABLE EQUATIONS Chapter 1. Introduction Chapter 2. Gemeral Remarks on Solutions Chapter 3. Families of Curves. Orthogonal Trajectories Chapter 4. Growth, Decay, Chemical Reactions, and Mixing Chapter 5. Falling Bodies and Other Motion Problems Chapter 6. The Brachistochrone. Fermat and the Bernoullis PART 2 FIRST ORDER EQUATIONS Chapter 7. Homogeneous Equations Chapter 8. Exact Equations Chapter 9. Integrating Factors Chapter 10. Linear Equations Chapter 11. Reduction of Order Chapter 12. The Hanging Chain. Pursuit Curves Chapter 13. Simple Electric Circuits PART 3 SECOND ORDER LINEAR EQUATIONS Chapter 14. Introduction Chapter 15. The General Solution of the Homogeneous Equation Chapter 16. The Use of a Known Solution to Find Another Chapter 17. The Homogeneous Equation with Constant Coefficients Chapter 18. The Method of Undetermined Coefficients Chapter 19. The Method of Variation and Parameters Chapter 20. Vibrations in Mechanical and Electrical Systems Chapter 21. Newton's Law of Gravitation and the Motions of the Planets Chapter 22. Higher Order Linear Equations. Coupled Harmonic Oscillators Chapter 23. Operator Methods for Finding Particular Solutions Appendix A. Euler Appendix B. Newton PART 4 QUALITATIVE PROPERTIES OF SOLUTIONS Chapter 24. Oscillations and the Sturm Separation Theorem Chapter 25. The Sturm Comparison Theorem PART 5 POWER SERIES SOLUTIONS AND SPECIAL FUNCTIONS Chapter 26. Introduction. A Review of Power Series Chapter 27. Series Solutions of First Order Equations Chapter 28. Second Order Linear Equations. Ordinary Points Chapter 29. Regular Singular Points Chapter 30. Regular Singular Points (Continued) Chapter 31. Gauss's Hypergeometric Equation Chapter 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 PART 6 FOURIER SERIES AND ORTHOGONAL FUNCTIONS Chapter 33. The Fourier Coefficients Chapter 34. The Problem of Convergence Chapter 35. Even and Odd Functions. Cosine and Sine Series Chapter 36. Extension to Arbitrary Intervals Chapter 37. Orthogonal Functions Chapter 38. The Mean Convergence of Fourier Series Appendix A. A Pointwise Convergence Theorem PART 7 PARTIAL DIFFERENTIAL EQUATIONS AND BOUNDARY VALUE PROBLEMS Chapter 39. Introduction. Historical Remarks Chapter 40. Eigenvalues, Eigenfunctions, and the Vibrating String Chapter 41. The Heat Equation Chapter 42. The Dirichlet Problem for a Circle. Poisson's Integral Chapter 43. Sturm-Liouville Problems Appendix A. The Existence of Eigenvalues and Eigenfunctions PART 8 SOME SPECIAL FUNCTIONS OF MATHEMATICAL PHYSICS Chapter 44. Legendre Polynomials Chapter 45. Properties of Legendre Polynomials Chapter 46. Bessel Functions. The Gamma Function Chapter 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 PART 9 LAPLACE TRANSFORMS Chapter 48. Introduction Chapter 49. A Few Remarks on the Theory Chapter 50. Applications to Differential Equations Chapter 51. Derivatives and Integrals of Laplace Transforms Chapter 52. Convolutions and Abel's Mechanical Problem Chapter 53. More about Convolutions. The Unit Step and Impulse Functions Appendix A. Laplace Appendix B. Abel PART 10 SYSTEMS OF FIRST ORDER EQUATIONS Chapter 54. General Remarks on Systems Chapter 55. Linear Systems Chapter 56. Homogeneous Linear Systems with Constant Coefficients Chapter 57. Nonlinear Systems. Volterra's Prey-Predator Equations PART 11 NONLINEAR EQUATIONS Chapter 58. Autonomous Systems. The Phase Plane and Its Phenomena Chapter 59. Types of Critical Points. Stability Chapter 60. Critical Points and Stability for Linear Systems Chapter 61. Stability by Liapunov's Direct Method Chapter 62. Simple Critical Points of Nonlinear Systems Chapter 63. Nonlinear Mechanics. Conservative Systems Chapter 64. Periodic Solutions. The Poincare-Bendixson Theorem Appendix A. Poincare Appendix B. Proof of Lienard?s Theorem PART 12 THE CALCULUS OF VARIATIONS Chapter 65. Introduction. Some Typical Problems of the Subject Chapter 66. Euler's Differential Equation for an Extremal Chapter 67. Isoperimetric problems Appendix A. Lagrange Appendix B. Hamilton's Principle and Its Implications PART 13 THE EXISTENCE AND UNIQUENESS OF SOLUTIONS Chapter 68. The Method of Successive Approximations Chapter 69. Picard's Theorem Chapter 70. Systems. The Second Order Linear Equation PART 14 NUMERICAL METHODS Chapter 71. Introduction Chapter 72. The Method of Euler Chapter 73. Errors Chapter 74. An Improvement to Euler Chapter 75. Higher-Order Methods Chapter 76. Systems Numerical Tables Answers Index