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Searching... | 30000010215843 | QA614.8 C35 2008 | Open Access Book | Book | Searching... |
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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
Preface | p. ix |
Chapter 1 First-Order Differential Equations and Their Applications | p. 1 |
1.1 Introduction to Ordinary Differential Equations | p. 1 |
1.2 The Definite Integral and the Initial Value Problem | p. 4 |
1.2.1 The Initial Value Problem and the Indefinite Integral | p. 5 |
1.2.2 The Initial Value Problem and the Definite Integral | p. 6 |
1.2.3 Mechanics I: Elementary Motion of a Particle with Gravity Only | p. 8 |
1.3 First-Order Separable Differential Equations | p. 13 |
1.3.1 Using Definite Integrals for Separable Differential Equations | p. 16 |
1.4 Direction Fields | p. 19 |
1.4.1 Existence and Uniqueness | p. 25 |
1.5 Euler's Numerical Method (optional) | p. 31 |
1.6 First-Order Linear Differential Equations | p. 37 |
1.6.1 Form of the General Solution | p. 37 |
1.6.2 Solutions of Homogeneous First-Order Linear Differential Equations | p. 39 |
1.6.3 Integrating Factors for First-Order Linear Differential Equations | p. 42 |
1.7 Linear First-Order Differential Equations with Constant Coefficients and Constant Input | p. 48 |
1.7.1 Homogeneous Linear Differential Equations with Constant Coefficients | p. 48 |
1.7.2 Constant Coefficient Linear Differential Equations with Constant Input | p. 50 |
1.7.3 Constant Coefficient Differential Equations with Exponential Input | p. 52 |
1.7.4 Constant Coefficient Differential Equations with Discontinuous Input | p. 52 |
1.8 Growth and Decay Problems | p. 59 |
1.8.1 A First Model of Population Growth | p. 59 |
1.8.2 Radioactive Decay | p. 65 |
1.8.3 Thermal Cooling | p. 68 |
1.9 Mixture Problems | p. 74 |
1.9.1 Mixture Problems with a Fixed Volume | p. 74 |
1.9.2 Mixture Problems with Variable Volumes | p. 77 |
1.10 Electronic Circuits | p. 82 |
1.11 Mechanics II: Including Air Resistance | p. 88 |
1.12 Orthogonal Trajectories (optional) | p. 92 |
Chapter 2 Linear Second- and Higher-Order Differential Equations | p. 96 |
2.1 General Solution of Second-Order Linear Differential Equations | p. 96 |
2.2 Initial Value Problem (for Homogeneous Equations) | p. 100 |
2.3 Reduction of Order | p. 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 Response | p. 124 |
2.5.1 Formulation of Equations | p. 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 Coefficients | p. 142 |
2.7 Mechanical Vibrations II: Forced Response | p. 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 Circuits | p. 174 |
2.9 Euler Equation | p. 179 |
2.10 Variation of Parameters (Second-Order) | p. 185 |
2.11 Variation of Parameters (nth-Order) | p. 193 |
Chapter 3 The Laplace Transform | p. 197 |
3.1 Definition and Basic Properties | p. 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 Equations | p. 225 |
3.4 Discontinuous Forcing Functions | p. 234 |
3.4.1 Solution of Differential Equations | p. 239 |
3.5 Periodic Functions | p. 248 |
3.6 Integrals and the Convolution Theorem | p. 253 |
3.6.1 Derivation of the Convolution Theorem (optional) | p. 256 |
3.7 Impulses and Distributions | p. 260 |
Chapter 4 An Introduction to Linear Systems of Differential Equations and Their Phase Plane | p. 265 |
4.1 Introduction | p. 265 |
4.2 Introduction to Linear Systems of Differential Equations | p. 268 |
4.2.1 Solving Linear Systems Using Eigenvalues and Eigenvectors of the Matrix | p. 269 |
4.2.2 Solving Linear Systems if the Eigenvalues are Real and Unequal | p. 272 |
4.2.3 Finding General Solutions of Linear Systems in the Case of Complex Eigenvalues | p. 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) Roots | p. 281 |
4.2.6 Eigenvalues and Trace and Determinant (optional) | p. 283 |
4.3 The Phase Plane for Linear Systems of Differential Equations | p. 287 |
4.3.1 Introduction to the Phase Plane for Linear Systems of Differential Equations | p. 287 |
4.3.2 Phase Plane for Linear Systems of Differential Equations | p. 295 |
4.3.3 Real Eigenvalues | p. 296 |
4.3.4 Complex Eigenvalues | p. 304 |
4.3.5 General Theorems | p. 310 |
Chapter 5 Mostly Nonlinear First-Order Differential Equations | p. 315 |
5.1 First-Order Differential Equations | p. 315 |
5.2 Equilibria and Stability | p. 316 |
5.2.1 Equilibrium | p. 316 |
5.2.2 Stability | p. 317 |
5.2.3 Review of Linearization | p. 318 |
5.2.4 Linear Stability Analysis | p. 318 |
5.3 One-Dimensional Phase Lines | p. 322 |
5.4 Application to Population Dynamics: The Logistic Equation | p. 327 |
Chapter 6 Nonlinear Systems of Differential Equations in the Plane | p. 332 |
6.1 Introduction | p. 332 |
6.2 Equilibria of Nonlinear Systems, Linear Stability Analysis of Equilibrium, and the Phase Plane | p. 335 |
6.2.1 Linear Stability Analysis and the Phase Plane | p. 336 |
6.2.2 Nonlinear Systems: Summary, Philosophy, Phase Plane, Direction Field, Nullclines | p. 341 |
6.3 Population Models | p. 349 |
6.3.1 Two Competing Species | p. 350 |
6.3.2 Predator-Prey Population Models | p. 356 |
6.4 Mechanical Systems | p. 363 |
6.4.1 Nonlinear Pendulum | p. 363 |
6.4.2 Linearized Pendulum | p. 364 |
6.4.3 Conservative Systems and the Energy Integral | p. 364 |
6.4.4 The Phase Plane and the Potential | p. 367 |
Answers to Odd-Numbered Exercises | p. 379 |
Index | p. 429 |