4. The Package ODE.m: 4.1 Getting Started with ODE; 4.2 Features of ODE; 4.3 Plotting with ODE; 4.4 First-Order Linear Equations via ODE; 4.5 Separable Equations via ODE; 4.6 First-Order Equations with Integrating Factors via ODE; 4.7 First-Order Homogeneous Equations via ODE; 4.8 Bernoulli Equations via ODE; 4.9 Clairaut and Lagrange Equations via ODE; 4.10 Nonelementary Integrals; 4.11 Using ODE to Define New Functions; 4.12 Riccati Equations

5. Existence and Uniqueness of Solutions of First-Order Differential Equations: 5.1 The Existence and Uniqueness Theorem; 5.2 Explosions and a Criterion for Global Existence; 5.3 Picard Iteration; 5.4 Proofs of Existence Theorems; 5.5 Direction Fields and Differential Equations; 5.6 Stability Analysis of Nonlinear First-Order Equations

6. Applications of First-Order Equations I: 6.1 Population Models with Constant Growth Rate; 6.2 Population Models with Variable Growth Rate; 6.3 Logistic Model of Population Growth; 6.4 Population Growth with Harvesting; 6.5 Population Models for the United States; 6.6 Temperature Equalization Models

7. Applications of First-Order Equations II: 7.1 Elementary Mechanics; 7.2 Rocket Propulsion; 7.3 Electrical Circuits; 7.4 Mixing Problems; 7.5 Pursuit Curves

8. Second-Order Linear Differential Equations: 8.1 General Forms and Examples; 8.2 Existence and Uniqueness Theory; 8.3 Fundamental Sets of Solutions to the Homogeneous Equation; 8.4 The Wronskian; 8.5 Linear Independence and the Wronskian; 8.6 Reduction of Order; 8.7 Equations with Given Solutions

9. Second-Order Linear Differential Equations with Constant Coefficients: 9.1 Constant-Coefficient Second-Order Homogeneous Equations; 9.2 Complex Constant-Coefficient Second-Order Homogeneous Equations; 9.3 The Method of Undetermined Coefficients; 9.4 The Method of Variation of Parameters

10. Using ODE to Solve Second-Order Linear Differential Equations: 10.1 Using ODE to Solve Second-Order Constant-Coefficient Equations; 10.2 Details of ODE for Second-Order Constant-Coefficient Equations; 10.3 Reduction of Order and Trial Solutions via ODE; 10.4 Equations with Given Solutions via ODE

11. Applications of Linear Second-Order Equations: 11.1 Mass-Spring Systems; 11.2 Forced Vibrations of Mass-Spring Systems; 11.3 Electrical Circuits; 11.4 Sound

12. Higher-Order Linear Differential Equations: 12.1 General Forms; 12.2 Constant-Coefficient Higher-Order Homogeneous Equations; 12.3 Variation of Parameters for Higher-Order Equations; 12.4 Higher-Order Differential Equations via ODE; 12.5 Seminumerical Solutions of Higher-Order Constant-Coefficient Equations

13. Numerical Solutions of Differential Equations: 13.1 The Euler Method; 13.2 The Heun Method; 13.3 The Runge-Kutta Method; 13.4 Solving Differential Equations Numerically with ODE; 13.5 ODE's Implementation of Numerical Methods; 13.6 Using NDSolve; 13.7 Adaptive Step Size and Error Control; 13.8 The Numerov Method

14. The Laplace Transform: 14.1 Definition and Properties of the Laplace Transform; 14.2 Piecewise Continuous Functions; 14.3 Using the Laplace Transform to Solve Initial Value Problems; 14.4 The Gamma Function; 14.5 Computation of Laplace Transforms; 14.6 Step Functions; 14.7 Second-Order Equations with Piecewise Continuous Forcing Functions; 14.8 Impulse Functions; 14.9 Convolution; 14.10 Laplace Transforms via ODE

15. Systems of Linear Differential Equations: 15.1 Notation and Definitions for Systems; 15.2 Existence and Uniqueness Theorems for Systems; 15.3 Solution of Upper Triangular Systems by Elimination; 15.4 Homogeneous Linear Systems; 15.5 Constant-Coefficient Homogeneous Systems; 15.6 The Method of Undetermined Coefficients for Systems; 15.7 The Method of Variation of Parameters for Systems; 15.8 Solving Systems Using the Laplace Transform

16. Phase Portraits of Linear Systems: 16.1 Phase Portraits of Two-Dimensional Linear Systems; 16.2 Using ODE to Solve Linear Systems; 16.3 Phase Portraits of Two-Dimensional Linear Systems via ODE

17. Stability of Nonlinear Systems: 17.1 Curves; 17.2 Autonomous Systems; 17.3 Critical Points of Systems of Differential Equations; 17.4 Stability and Asymptotic Stability of Nonlinear Systems; 17.5 Stability by Linearized Approximation; 17.6 Lyapunov Stability Theory

18. Applications of Linear Systems: 18.1 Coupled Systems of Oscillators; 18.2 Electrical Circuits; 18.3 Markov Chains

19. Applications of Nonlinear Systems: 19.1 Numerical Solutions of Systems of Differential Equations; 19.2 Predator-Prey Modeling; 19.3 The Van Der Pol Equation; 19.4 The Simple Pendulum; 19.5 The Fundamental Theorem of Plane Curves

20. Power Series Solutions of Second-Order Equations: 20.1 Review of Power Series; 20.2 Power Series via Mathematica; 20.3 Power Series Solutions about an Ordinary Point; 20.4 The Airy Equation; 20.5 The Legendre Equation; 20.6 Convergence of Series Solutions; 20.7 Series Solutions of Differential Equations Using ODE

21. Frobenius Solutions of Second-Order Equations: 21.1 Solutions about a Regular Singular Point; 21.2 The Cauchy-Euler Equation; 21.3 Method of Frobenius: The First Solution; 21.4 Bessel Functions I; 21.5 Method of Frobenius: The Second Solution; 21.6 Bessel Functions II; 21.7 Bessel Functions via Mathematica; 21.8 An Aging Spring; 21.9 The Hypergeometric Equation

A. Appendix: Review of Linear Algebra and Matrix Theory: A.1 Vector and Matrix Notation; A.2 Determinants and Inverses; A.3 Systems of Linear Equations and Determinants; A.4 Eigenvalues and Eigenvectors; A.5 The Exponential of a Matrix; A.6 Abstract Vector Spaces; A.7 Vectors and Matrices with Mathematica; A.8 Solving Equations with Mathematica; A.9 Eigenvalues and Eigenvectors with Mathematica

B. Appendix: Systems of Units: Answers; Bibliography; General Index; Name Index; Miniprogram and Mathematica Index

Table of Contents of the Text