Differential Equations Graphics, Models, Data

Viewing stained glass from different angles or in various lights is necessary to discover its many qualities. Likewise, viewing solutions of differential equations from several points of view is essential to fully understand their behavior. Lomen and Lovelock provide an active environment for students to explore differential equations by using analytical, numerical, graphical, and descriptive techniques, and for students to use ODEs as a natural tool for modeling many interesting processes in science and engineering.

David O. Lomen is the author of Differential Equations: Graphics, Models, Data, published by Wiley. David Lovelock is a British theoretical physicist and mathematician. He is known for Lovelock theory of gravity and the Lovelock's theorem.

PREFACE

v

CHAPTER 1 BASIC CONCEPTS

1

(26)

1.1 Simple Differential Equations and Explicit Solutions

1

(6)

1.2 Graphical Solutions Using Calculus

7

(4)

1.3 Slope Fields and Isoclines

11

(11)

1.4 Functions and Power Series Expansions

22

(5)

CHAPTER 2 AUTONOMOUS DIFFERENTIAL EQUATIONS

27

(54)

2.1 Autonomous Equations

28

(8)

2.2 Simple Models

36

(6)

2.3 The Logistic Equation

42

(11)

2.4 Existence and Uniqueness of Solutions, and Words of Caution

53

(12)

2.5 Qualitative Behavior of Solutions Using Phases Lines

65

(10)

2.6 Bifurcation Diagrams

75

(6)

CHAPTER 3 FIRST ORDER DIFFERENTIAL EQUATIONS -- QUALITATIVE AND QUANTITATIVE ASPECTS

81

(38)

3.1 Graphical Solutions Using Calculus

81

(13)

3.2 Symmetry of Slope Fields

94

(3)

3.3 Numerical Solutions and Chaos

97

(15)

3.4 Comparing Solutions of Differential Equations

112

(3)

3.5 Finding Power Series Solutions

115

(4)

CHAPTER 4 MODELS AND APPLICATIONS LEADING TO NEW TECHNIQUES

119

(64)

4.1 Solving Separable Differential Equations

119

(17)

4.2 Solving Differential Equations with Homogeneous Coefficients

136

(9)

4.3 Models: Deriving Differential Equations From Data

145

(13)

4.4 Models: Objects in Motion

158

(9)

4.5 Application: Orthogonal Trajectories

167

(8)

4.6 Piecing Together Differential Equations

175

(8)

CHAPTER 5 FIRST ORDER LINEAR DIFFERENTIAL EQUATIONS AND MODELS

183

(40)

5.1 Solving Linear Differential Equations

183

(13)

5.2 Models That Use Linear Equations

196

(15)

5.3 Models That Use Bernoulli's Equation

211

(12)

CHAPTER 6 INTERPLAY BETWEEN FIRST ORDER SYSTEMS AND SECOND ORDER EQUATIONS

223

(56)

6.1 Simple Models

223

(8)

6.2 How First Order Systems and Second Order Equations Are Related

231

(10)

6.3 Second Order Linear Differential Equations with Constant Coefficients

241

(9)

6.4 Modeling Physical Situations

250

(9)

6.5 Interpreting the Phase Plane

259

(7)

6.6 How Explicit Solutions Are Related to Orbits

266

(6)

6.7 The Motion of a Nonlinear Pendulum

272

(7)

CHAPTER 7 SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS WITH FORCING FUNCTIONS

279

(36)

7.1 The General Solution

279

(4)

7.2 Finding Solutions by the Method of Undetermined Coefficients

283

(11)

7.3 Application and Models

294

(21)

CHAPTER 8 SECOND ORDER LINEAR DIFFERENTIAL EQUATIONS -- QUALITATIVE AND QUANTITATIVE ASPECTS

315

(64)

8.1 Qualitative Behavior of Solutions

315

(8)

8.2 Finding Solutions by Reduction of Order

323

(7)

8.3 Finding Solutions by Variation of Parameters

330

(6)

8.4 The Importance of Linear Independence and Dependence

336

(7)

8.5 Solving Cauchy -- Euler Equations

343

(7)

8.6 Boundary Value Problems and the Shooting Method

350

(5)

8.7 Solving Higher Order Homogeneous Differential Equations

355

(13)

8.8 Solving Higher Order Nonhomogeneous Differential Equations

368

(11)

CHAPTER 9 LINEAR AUTONOMOUS SYSTEMS

379

9.1 Solving Linear Autonomous Systems

379

(9)

9.2 Classification of Solutions via Stability

388

(15)

9.3 When Do Straight -- Line Orbits Exist?

403

(5)

9.4 Qualitative Behavior Using Nullclines

408

(5)

9.5 Matrix Formulation of Solutions

413

(17)

9.6 Compartmental Models

430

CHAPTER 10 NONLINEAR AUTONOMOUS SYSTEMS

315

10.1 Introduction to Nonlinear Autonomous Systems

437

(11)

10.2 Qualitative Behavior Using Nullcline Analysis

448

(8)

10.3 Qualitative Behavior Using Linearization

456

(11)

10.4 Models Involving Nonlinear Autonomous Equations

467

(18)

10.5 Bungee Jumping

485

(7)

10.6 Linear Versus Nonlinear Differential Equations

492

(6)

10.7 Autonomous Versus Nonautonomous Differential Equations

498

(7)

CHAPTER 11 USING LAPLACE TRANSFORMS

505

(70)

11.1 Motivation

505

(11)

11.2 Constructing New Laplace Transforms from Old

516

(7)

11.3 The Inverse Laplace Transform and the Convolution Theorem

523

(8)

11.4 Functions That Jump

531

(14)

11.5 Models Involving First Order Linear Differential Equations

545

(8)

11.6 Models Involving Higher Order Linear Differential Equations

553

(7)

11.7 Applications to Systems of Linear Differential Equations

560

(6)

11.8 When Do Laplace Transforms Exist?

566

(9)

CHAPTER 12 USING POWER SERIES

575

(56)

12.1 Solutions Using Taylor Series

576

(5)

12.2 Solutions Using Power Series

581

(18)

12.3 What To Do When Power Series Fail

599

(10)

12.4 Solutions Using the Method of Frobenius

609

(22)

APPENDICES

631

(28)

A.1 Background Material

631

(6)

A.2 Partial Fractions

637

(3)

A.3 Infinite Series, Power Series, and Taylor Series

640

(2)

A.4 Complex Numbers

642

(3)

A.5 Elementary Matrix Operations

645

(6)

A.6 Least Squares Approximation

651

(3)

A.7 Proofs of the Oscillation Theorems

654

(5)

ANSWERS

659

(13)

INDEX

672

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Differential Equations Graphics, Models, Data

Summary

Author Biography

Table of Contents

Supplemental Materials

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Differential Equations Graphics, Models, Data

Summary

Author Biography

Table of Contents

Supplemental Materials

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