Book:George F. Simmons/Differential Equations

Subject Matter

 * Differential Equations

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