Introduction to Differential Equations Using Sage

by David Joyner, Marshall Hampton

Overview

Differential equations can be taught using Sage as an inventive new approach.

David Joyner and Marshall Hampton's lucid textbook explains differential equations using the free and open-source mathematical software Sage.

Since its release in 2005, Sage has acquired a substantial following among mathematicians, but its first user was Joyner, who is credited with helping famed mathematician William Stein turn the program into a usable and popular choice.

Introduction to Differential Equations Using Sage extends Stein's work by creating a classroom tool that allows both differential equations and Sage to be taught concurrently. It's a creative and forward-thinking approach to math instruction.

Topics include:

• First-Order Differential Equations
• Incorporation of Newtonian Mechanics
• Second-Order Differential Equations
• The Annihilator Method
• Using Linear Algebra with Differential Equations
• Nonlinear Systems
• Partial Differential Equations
• Romeo and Juliet

Product Details

ISBN-13:	9781421406374
Publisher:	Johns Hopkins University Press
Publication date:	09/01/2012
Pages:	280
Product dimensions:	10.00(w) x 7.20(h) x 1.00(d)
Age Range:	18 Years

About the Author

David Joyner is a professor in the Mathematics Department at the U.S. Naval Academy. He is the author of Adventures in Group Theory: Rubik's Cube, Merlin's Machine, and Other Mathematical Toys, also published by Johns Hopkins.

Table of Contents

Preface xi

Acknowledgments xiii

1 First-order differential equations 1

1.1 Introduction to DEs 1

1.2 Initial value problems 10

1.3 Existence of solutions to ODEs 15

1.3.1 First-order ODEs 15

1.3.2 Second-order homogeneous ODEs 19

1.4 First-order ODEs: Separable and linear cases 22

1.4.1 Separable DEs 22

1.4.2 Autonomous ODEs 26

1.4.3 Substitution methods 29

1.4.4 Linear first-order ODEs 29

1.5 Isoclines and direction fields 33

1.6 Numerical solutions: Euler's and improved Euler's method 38

1.6.1 Euler's method 38

1.6.2 Improved Euler's method 41

1.6.3 Euler's method for systems and higher-order DEs 44

1.7 Numerical solutions II: Runge-Kutta and other methods 47

1.7.1 Fourth-order Runge-Kutta method 47

1.7.2 Multistep methods: Adams-Bashforth 49

1.7.3 Adaptive step size 49

1.8 Newtonian mechanics 51

1.9 Application to mixing problems 56

1.10 Application to cooling problems 60

2 Second-order differential equations 65

2.1 Linear differential equations 65

2.1.1 Solving homogeneous constant-coefficient ODEs 65

2.2 Linear differential equations, revisited 69

2.3 Linear differential equations, continued 74

2.4 Undetermined coefficients method 79

2.4.1 Simple case 80

2.4.2 Nonsimple case 82

2.5 Annihilator method 87

2.6 Variation of parameters 89

2.6.1 The Leibniz rule 89

2.6.2 The method 90

2.7 Applications of DEs: Spring problems 92

2.7.1 Introduction: Simple harmonic case 92

2.7.2 Simple harmonic case 94

2.7.3 Free damped motion 97

2.7.4 Spring-mass systems with an external force 101

2.8 Applications to simple LRC circuits 105

2.9 The power series method 110

2.9.1 Part 1 110

2.9.2 Part 2 118

2.10 The Laplace transform method 121

2.10.1 Part 1 121

2.10.2 Part 2 129

2.10.3 Part 3 138

3 Matrix theory and systems of DEs 141

3.1 Quick survey of linear algebra 141

3.1.1 Matrix arithmetic 141

3.2 Row reduction and solving systems of equations 145

3.2.1 The Gauss elimination game 145

3.2.2 Solving systems using inverses 149

3.2.3 Computing inverses using row reduction 149

3.2.4 Solving higher-dimensional linear systems 155

3.2.5 Determinants 156

3.2.6 Elementary matrices and computation of determinants 159

3.2.7 Vector spaces 161

3.2.8 Bases, dimension, linear independence, and span 163

3.3 Application: Solving systems of DEs 166

3.3.1 Modeling battles using Lanchester's equations 170

3.3.2 Romeo and Juliet 178

3.3.3 Electrical networks using Laplace transforms 183

3.4 Eigenvalue method for systems of DEs 188

3.4.1 Motivation 188

3.4.2 Computing eigenvalues 191

3.4.3 The eigenvalue method 195

3.4.4 Examples of the eigenvalue method 196

3.5 Introduction to variation of parameters for systems 201

3.5.1 Motivation 201

3.5.2 The method 203

3.6 Nonlinear systems 207

3.6.1 Linearizing near equilibria 208

3.6.2 The nonlinear pendulum 209

3.6.3 The Lorenz equations 211

3.6.4 Zombies attack 212

4 Introduction to partial differential equations 219

4.1 Introduction to separation of variables 219

4.1.1 The transport or advection equation 220

4.1.2 The heat equation 223

4.2 The method of superposition 225

4.3 Fourier, sine, and cosine series 228

4.3.1 Brief history 228

4.3.2 Motivation 228

4.3.3 Definitions 229

4.4 The heat equation 237

4.4.1 Method for zero ends 238

4.4.2 Method for insulated ends 240

4.4.3 Explanation via separation of variables 243

4.5 The wave equation in one dimension 246

4.5.1 Method 247

4.6 The Schrödinger equation 250

4.6.1 Method 251

Bibliography 255

Index 261

Editorial Reviews

This book, with its many practice examples, would be ideal for those in the first year of a mathematics degree course and for those studying for working in physics or any area requiring a good knowledge of differential equations.
—Adrian Hamilton, Institute of Mathematics and its Applications

. . . [Introduction to Differential Equations Using Sage] provides a nice mix of theory and symbolic computation. The pedagogy is excellent and the exercises are models of their kind.
—Gazette of the Australian Mathematical Society

From the Publisher

. . . [Introduction to Differential Equations Using Sage] provides a nice mix of theory and symbolic computation. The pedagogy is excellent and the exercises are models of their kind.

Gazette of the Australian Mathematical Society

This book, with its many practice examples, would be ideal for those in the first year of a mathematics degree course and for those studying for working in physics or any area requiring a good knowledge of differential equations.

Institute of Mathematics and its Applications - Adrian Hamilton