Mathematica

Introduction to Ordinary Differential Equations with Mathematica ®

Authors

Alfred Gray
Department of Mathematics
University of Maryland
College Park, Maryland

Michael Mezzino
Department of Mathematics
University of Houston - Clear Lake
Houston, Texas

Mark Pinksy
Department of Mathematics
Northwestern University
Evanston, Illinois

Overview
This book/CD-ROM combination provides a traditional treatment of elementary ordinary differential equations while introducing computer-assisted methods now available with Mathematica. Classical solution methods are presented in parallel with those in Mathematica. Models are developed from calssical physics, popluation biology, electrical circuits, and elementary mechanics.

Included on the multi-platform CD-ROM are Mathematica notebooks, movies of phase plane and phase space portraits, and a unique piece of software, ODE.m, which contains a set of tools for working with ordinary differential equations solvers by providing dozens of additional method.

A prerequisite for using these materials is a course in the calculus of one variable, although multi-variable calculus and linear algebra are recommended. The text covers standard topics in first and second order differential equations, power series solutions, first-order systems, Laplace transforms, numerical methods, and stability of nonlinear systems. The text includes more than 650 exercises and hundreds of worked examples throughout. A brief introduction to using Mathematica is provided both in the book and on the CD-ROM.

1997/890 pages/Hardcover Book/CD-ROM Package
ISBN 0-387-94481-8

The solutions manual to the text is available for instructor use.
1998/530 pages/Softcover Manual
ISBN 0-387-98232-9

System Requirements

The cross-platform CD-ROM is compatible with Macintosh, NeXT, Windows, and most UNIX computer systems. Mathematica 2.2 or higher is recommended to fully utilize CD contents.

MathReader is a freely available program that will allow you to read (and print) any Mathematica notebook on most computer platforms.

Contents

Table of Contents


Preface
1. Basic Concepts
1.1 The Notion of a Differential Equation
1.2 Sources of Differential Equations
1.3 Solving Differential Equations

2. Using Mathematica
2.1 Getting Started with Mathematica
2.2 Mathematica Notation versus Ordinary Mathematical Notation
2.3 Plotting in Mathematica

3. First-Order Differential Equations
3.1 Introduction to First-Order Equations
3.2 First-Order Linear Equations
3.3 Separable Equations
3.4 Exact Equations and Integrating Factors
3.5 Homogeneous First-Order Equations
3.6 Bernoulli Equations

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 Application of First-Order Equations to Elementary Mechanics
7.2 Rocket Propulsion
7.3 Application of First-Order Equations to 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.m to Solve Second-Order Linear Differential Equations
10.1 Using ODE.m to Solve Second-Order Constant-Coefficient Equations
10.2 Details of ODE.m for Second-Order Constant-Coefficient Equations
10.3 Reduction of Order and Trial Solutions via ODE
10.4 Equations with Given Solutions via ODE.m

11. Applications of Linear Second-Order Equations
11.1 Mass-Spring Systems
11.2 Forced Vibrations of Mass-Spring Systems
11.3 Applications of Second-Order Equations to 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 Mathematica

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 Applications to Electrical Circuits
18.3 Applications to 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

About the Authors

 The author-team combines over 60 years of teaching experience in the development of these course materials and the accompanying software that comprise this unique teaching package. Alfred Gray Michael Mezzino and Mark Pinsky are well-known educators and authors in the mathematics community.

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Secaucus, NJ 07096-2485
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