Cover image for Introduction to differential equations with dynamical systems
Title:
Introduction to differential equations with dynamical systems
Personal Author:
Publication Information:
New Jersey : Princeton University Press, 2008
Physical Description:
xiii, 430 p. : ill. ; 26 cm.
ISBN:
9780691124742

Available:*

Library
Item Barcode
Call Number
Material Type
Item Category 1
Status
Searching...
30000010215843 QA614.8 C35 2008 Open Access Book Book
Searching...

On Order

Summary

Summary

Many textbooks on differential equations are written to be interesting to the teacher rather than the student. Introduction to Differential Equations with Dynamical Systems is directed toward students. This concise and up-to-date textbook addresses the challenges that undergraduate mathematics, engineering, and science students experience during a first course on differential equations. And, while covering all the standard parts of the subject, the book emphasizes linear constant coefficient equations and applications, including the topics essential to engineering students. Stephen Campbell and Richard Haberman--using carefully worded derivations, elementary explanations, and examples, exercises, and figures rather than theorems and proofs--have written a book that makes learning and teaching differential equations easier and more relevant. The book also presents elementary dynamical systems in a unique and flexible way that is suitable for all courses, regardless of length.


Author Notes

Stephen L. Campbell is professor of mathematics and director of the graduate program in mathematics at North Carolina State University.


Reviews 1

Choice Review

Campbell (North Carolina State Univ.) and Haberman (Southern Methodist Univ.) have written a relatively concise book covering the traditional topics addressed in a first course on ordinary differential equations (ODEs). There is a considerable amount of emphasis on applications throughout the volume, which is well written, if fairly traditional. The topics include first, second, and higher order linear ODEs; linear systems, phase planes, and the roles of eigenvalues and associated eigenvectors; first order nonlinear ODEs and nonlinear planar systems; and Laplace transforms. The authors discuss various applications such as radioactive decay and thermal cooling; electrical circuits; damped oscillations, mass, spring, and pendulum systems; and predator-prey population models. Although there are no references (this is very standard material), the book provides 1,275 exercises (ranging from routine to challenging) and 165 examples. Summing Up: Recommended. Lower-division undergraduates. J. D. Fehribach Worcester Polytechnic Institute


Table of Contents

Prefacep. ix
Chapter 1 First-Order Differential Equations and Their Applicationsp. 1
1.1 Introduction to Ordinary Differential Equationsp. 1
1.2 The Definite Integral and the Initial Value Problemp. 4
1.2.1 The Initial Value Problem and the Indefinite Integralp. 5
1.2.2 The Initial Value Problem and the Definite Integralp. 6
1.2.3 Mechanics I: Elementary Motion of a Particle with Gravity Onlyp. 8
1.3 First-Order Separable Differential Equationsp. 13
1.3.1 Using Definite Integrals for Separable Differential Equationsp. 16
1.4 Direction Fieldsp. 19
1.4.1 Existence and Uniquenessp. 25
1.5 Euler's Numerical Method (optional)p. 31
1.6 First-Order Linear Differential Equationsp. 37
1.6.1 Form of the General Solutionp. 37
1.6.2 Solutions of Homogeneous First-Order Linear Differential Equationsp. 39
1.6.3 Integrating Factors for First-Order Linear Differential Equationsp. 42
1.7 Linear First-Order Differential Equations with Constant Coefficients and Constant Inputp. 48
1.7.1 Homogeneous Linear Differential Equations with Constant Coefficientsp. 48
1.7.2 Constant Coefficient Linear Differential Equations with Constant Inputp. 50
1.7.3 Constant Coefficient Differential Equations with Exponential Inputp. 52
1.7.4 Constant Coefficient Differential Equations with Discontinuous Inputp. 52
1.8 Growth and Decay Problemsp. 59
1.8.1 A First Model of Population Growthp. 59
1.8.2 Radioactive Decayp. 65
1.8.3 Thermal Coolingp. 68
1.9 Mixture Problemsp. 74
1.9.1 Mixture Problems with a Fixed Volumep. 74
1.9.2 Mixture Problems with Variable Volumesp. 77
1.10 Electronic Circuitsp. 82
1.11 Mechanics II: Including Air Resistancep. 88
1.12 Orthogonal Trajectories (optional)p. 92
Chapter 2 Linear Second- and Higher-Order Differential Equationsp. 96
2.1 General Solution of Second-Order Linear Differential Equationsp. 96
2.2 Initial Value Problem (for Homogeneous Equations)p. 100
2.3 Reduction of Orderp. 107
2.4 Homogeneous Linear Constant Coefficient Differential Equations (Second Order)p. 112
2.4.1 Homogeneous Linear Constant Coefficient Differential Equations (nth-Order)p. 122
2.5 Mechanical Vibrations I: Formulation and Free Responsep. 124
2.5.1 Formulation of Equationsp. 124
2.5.2 Simple Harmonic Motion (No Damping, [delta] = 0)p. 128
2.5.3 Free Response with Friction ([delta] > 0)p. 135
2.6 The Method of Undetermined Coefficientsp. 142
2.7 Mechanical Vibrations II: Forced Responsep. 159
2.7.1 Friction is Absent ([delta] = 0)p. 159
2.7.2 Friction is Present ([delta] > 0) (Damped Forced Oscillations)p. 168
2.8 Linear Electric Circuitsp. 174
2.9 Euler Equationp. 179
2.10 Variation of Parameters (Second-Order)p. 185
2.11 Variation of Parameters (nth-Order)p. 193
Chapter 3 The Laplace Transformp. 197
3.1 Definition and Basic Propertiesp. 197
3.1.1 The Shifting Theorem (Multiplying by an Exponential)p. 205
3.1.2 Derivative Theorem (Multiplying by t)p. 210
3.2 Inverse Laplace Transforms (Roots, Quadratics, and Partial Fractions)p. 213
3.3 Initial Value Problems for Differential Equationsp. 225
3.4 Discontinuous Forcing Functionsp. 234
3.4.1 Solution of Differential Equationsp. 239
3.5 Periodic Functionsp. 248
3.6 Integrals and the Convolution Theoremp. 253
3.6.1 Derivation of the Convolution Theorem (optional)p. 256
3.7 Impulses and Distributionsp. 260
Chapter 4 An Introduction to Linear Systems of Differential Equations and Their Phase Planep. 265
4.1 Introductionp. 265
4.2 Introduction to Linear Systems of Differential Equationsp. 268
4.2.1 Solving Linear Systems Using Eigenvalues and Eigenvectors of the Matrixp. 269
4.2.2 Solving Linear Systems if the Eigenvalues are Real and Unequalp. 272
4.2.3 Finding General Solutions of Linear Systems in the Case of Complex Eigenvaluesp. 276
4.2.4 Special Systems with Complex Eigenvalues (optional)p. 279
4.2.5 General Solution of a Linear System if the Two Real Eigenvalues are Equal (Repeated) Rootsp. 281
4.2.6 Eigenvalues and Trace and Determinant (optional)p. 283
4.3 The Phase Plane for Linear Systems of Differential Equationsp. 287
4.3.1 Introduction to the Phase Plane for Linear Systems of Differential Equationsp. 287
4.3.2 Phase Plane for Linear Systems of Differential Equationsp. 295
4.3.3 Real Eigenvaluesp. 296
4.3.4 Complex Eigenvaluesp. 304
4.3.5 General Theoremsp. 310
Chapter 5 Mostly Nonlinear First-Order Differential Equationsp. 315
5.1 First-Order Differential Equationsp. 315
5.2 Equilibria and Stabilityp. 316
5.2.1 Equilibriump. 316
5.2.2 Stabilityp. 317
5.2.3 Review of Linearizationp. 318
5.2.4 Linear Stability Analysisp. 318
5.3 One-Dimensional Phase Linesp. 322
5.4 Application to Population Dynamics: The Logistic Equationp. 327
Chapter 6 Nonlinear Systems of Differential Equations in the Planep. 332
6.1 Introductionp. 332
6.2 Equilibria of Nonlinear Systems, Linear Stability Analysis of Equilibrium, and the Phase Planep. 335
6.2.1 Linear Stability Analysis and the Phase Planep. 336
6.2.2 Nonlinear Systems: Summary, Philosophy, Phase Plane, Direction Field, Nullclinesp. 341
6.3 Population Modelsp. 349
6.3.1 Two Competing Speciesp. 350
6.3.2 Predator-Prey Population Modelsp. 356
6.4 Mechanical Systemsp. 363
6.4.1 Nonlinear Pendulump. 363
6.4.2 Linearized Pendulump. 364
6.4.3 Conservative Systems and the Energy Integralp. 364
6.4.4 The Phase Plane and the Potentialp. 367
Answers to Odd-Numbered Exercisesp. 379
Indexp. 429