Differential Equations

by Polking, John; Boggess, Al; Arnold, David

• ISBN13: 9780131437388
• ISBN10: 0131437380
• Edition: 2nd
• Format: Hardcover
• Copyright: 2005-07-14
• Publisher: Pearson

Summary

Combining traditional material with a modern systems approach, this handbook provides a thorough introduction to differential equations, tempering its classic "pure math" approach with more practical applied aspects.Features up-to-date coverage of key topics such as first order equations, matrix algebra, systems, and phase plane portraits. Illustrates complex concepts through extensive detailed figures. Focuses on interpreting and solving problems through optional technology projects.For anyone interested in learning more about differential equations.

Table of Contents

Preface	p. ix
Introduction to Differential Equations	p. 1
Differential Equation Models	p. 2
The Derivative	p. 6
Integration	p. 10
First-Order Equations	p. 16
Differential Equations and Solutions	p. 16
Solutions to Separable Equations	p. 27
Models of Motion	p. 37
Linear Equations	p. 47
Mixing Problems	p. 56
Exact Differential Equations	p. 63
Existence and Uniqueness of Solutions	p. 77
Dependence of Solutions on Initial Conditions	p. 88
Autonomous Equations and Stability	p. 92
The Daredevil Skydiver	p. 102
Modeling and Applications	p. 104
Modeling Population Growth	p. 105
Models and the Real World	p. 117
Personal Finance	p. 121
Electrical Circuits	p. 128
The Spruce Budworm	p. 132
Social Security, Now or Later	p. 134
Second-Order Equations	p. 136
Definitions and Examples	p. 136
Second-Order Equations and Systems	p. 146
Linear, Homogeneous Equations with Constant Coefficients	p. 150
Harmonic Motion	p. 156
Inhomogeneous Equations; the Method of Undetermined Coefficients	p. 164
Variation of Parameters	p. 173
Forced Harmonic Motion	p. 177
Nonlinear Oscillators	p. 187
The Laplace Transform	p. 189
The Definition of the Laplace Transform	p. 190
Basic Properties of the Laplace Transform	p. 197
The Inverse Laplace Transform	p. 203
Using the Laplace Transform to Solve Differential Equations	p. 209
Discontinuous Forcing Terms	p. 215
The Delta Function	p. 227
Convolutions	p. 233
Summary	p. 242
Forced Harmonic Oscillators	p. 243
Numerical Methods	p. 245
Euler's Method	p. 246
Runge-Kutta Methods	p. 255
Numerical Error Comparisons	p. 261
Practical Use of Solvers	p. 265
Numerical Error Comparison	p. 271
Matrix Algebra	p. 272
Vectors and Matrices	p. 272
Systems of Linear Equations with Two or Three Variables	p. 283
Solving Systems of Equations	p. 292
Homogeneous and Inhomogeneous Systems	p. 300
Bases of a Subspace	p. 307
Square Matrices	p. 317
Determinants	p. 322
An Introduction to Systems	p. 331
Definitions and Examples	p. 331
Geometric Interpretation of Solutions	p. 338
Qualitative Analysis	p. 347
Linear Systems	p. 353
Properties of Linear Systems	p. 362
Long-Term Behavior of Solutions	p. 370
Linear Systems with Constant Coefficients	p. 372
Overview of the Technique	p. 372
Planar Systems	p. 378
Phase Plane Portraits	p. 392
The Trace-Determinant Plane	p. 402
Higher-Dimensional Systems	p. 407
The Exponential of a Matrix	p. 416
Qualitative Analysis of Linear Systems	p. 429
Higher-Order Linear Equations	p. 433
Inhomogeneous Linear Systems	p. 444
Phase Plane Portraits	p. 454
Oscillations of Linear Molecules	p. 454
Nonlinear Systems	p. 458
The Linearization of a Nonlinear System	p. 458
Long-Term Behavior of Solutions	p. 469
Invariant Sets and the Use of Nullclines	p. 475
Long-Term Behavior of Solutions to Planar Systems	p. 481
Conserved Quantities	p. 490
Nonlinear Mechanics	p. 495
The Method of Lyapunov	p. 510
Predator-Prey Systems	p. 517
Human Immune Response to Infectious Disease	p. 528
Analysis of Competing Species	p. 530
Series Solutions to Differential Equations	p. 532
Review of Power Series	p. 533
Series Solutions Near Ordinary Points	p. 543
Legendre's Equation	p. 555
Types of Singular Points-Euler's Equation	p. 560
Series Solutions Near Regular Singular Points	p. 566
Series Solutions Near Regular Singular Points-The General Case	p. 575
Bessel's Equation and Bessel Functions	p. 586
Complex Numbers and Matrices	p. 595
Answers to Odd-Numbered Problems	p. 1
Index	p. 1
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