DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTE

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 Poincaré-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