Differential equations, with applications and historical notes / George F. Simmons, with a new chapter on numerical methods by John S. Robertson.

Includes bibliographical references and index.

Preface to the Second EditionPreface to the First EditionSuggestions for the Instructor1 The Nature of Differential Equations. Separable Equations1. Introduction2. Gemeral Remarks on Solutions3. Families of Curves. Orthogonal Trajectories4. Growth, Decay, Chemical Reactions, and Mixing5. Falling Bodies and Other Motion Problems6. The Brachistochrone. Fermat and the Bernoullis2 First Order Equations7. Homogeneous Equations8. Exact Equations9. Integrating Factors10. Linear Equations11. Reduction of Order12. The Hanging Chain. Pursuit Curves13. Simple Electric Circuits3 Second Order Linear Equations14. Introduction15. The General Solution of the Homogeneous Equation16. The Use of a Known Solution to Find Another17. The Homogeneous Equation with Constant Coefficients18. The Method of Undetermined Coefficients19. The Method of Variation and Parameters20. Vibrations in Mechanical and Electrical Systems21. Newton's Law of Gravitation and the Motions of the Planets22. Higher Order Linear Equations. Coupled Harmonic Oscillators23. Operator Methods for Finding Particular SolutionsAppendix A. EulerAppendix B. Newton4 Qualitative Properties of Solutions24. Oscillations and the Sturm Separation Theorem25. The Sturm Comparison Theorem5 Power Series Solutions and Special Functions26. Introduction. A Review of Power Series27. Series Solutions of First Order Equations28. Second Order Linear Equations. Ordinary Points29. Regular Singular Points30. Regular Singular Points (Continued)31. Gauss's Hypergeometric Equation32. The Point at InfinityAppendix A. Two Convergence ProofsAppendix B. Hermite Polynomials and Quantum MechanicsAppendix C. GaussAppendix D. Chebyshev Polynomials and the Minimax PropertyAppendix E. Riemann's Equation6 Fourier Series and Orthogonal Functions33. The Fourier Coefficients34. The Problem of Convergence35. Even and Odd Functions. Cosine and Sine Series36. Extension to Arbitrary Intervals37. Orthogonal Functions38. The Mean Convergence of Fourier SeriesAppendix A. A Pointwise Convergence Theorem7 Partial Differential Equations and Boundary Value Problems39. Introduction. Historical Remarks40. Eigenvalues, Eigenfunctions, and the Vibrating String41. The Heat Equation42. The Dirichlet Problem for a Circle. Poisson's Integral43. Sturm-Liouville ProblemsAppendix A. The Existence of Eigenvalues and Eigenfunctions8 Some Special Functions of Mathematical Physics44. Legendre Polynomials45. Properties of Legendre Polynomials46. Bessel Functions. The Gamma Function47. Properties of Bessel functionsAppendix A. Legendre Polynomials and Potential TheoryAppendix B. Bessel Functions and the Vibrating MembraneAppendix C. Additional Properties of Bessel Functions9 Laplace Transforms48. Introduction49. A Few Remarks on the Theory50. Applications to Differential Equations51. Derivatives and Integrals of Laplace Transforms52. Convolutions and Abel's Mechanical Problem53. More about Convolutions. The Unit Step and Impulse FunctionsAppendix A. LaplaceAppendix B. Abel10 Systems of First Order Equations54. General Remarks on Systems55. Linear Systems56. Homogeneous Linear Systems with Constant Coefficients57. Nonlinear Systems. Volterra's Prey-Predator Equations11 Nonlinear Equations58. Autonomous Systems. The Phase Plane and Its Phenomena59. Types of Critical Points. Stability.60. Critical Points and Stability for Linear Systems61. Stability by Liapunov's Direct Method62. Simple Critical Points of Nonlinear Systems63. Nonlinear Mechanics. Conservative Systems64. Periodic Solutions. The Poincare-Bendixson TheoremAppendix A. PoincareAppendix B. Proof of Lienard's Theorem12 The Calculus of Variations65. Introduction. Some Typical Problems of the Subject66. Euler's Differential Equation for an Extremal67. Isoperimetric problemsAppendix A. LagrangeAppendix B. Hamilton's Principle and Its Implications13 The Existence and Uniqueness of Solutions68. The Method of Successive Approximations69. Picard's Theorem70. Systems. The Second Order Linear Equation14 Numerical Methods71. Introduction72. The Method of Euler73. Errors74. An Improvement to Euler75. Higher-Order Methods76. SystemsNumerical TablesAnswersIndex