A First Course in Differential Equations Third Edition
[Pages:380]J. David Logan Department of Mathematics University of Nebraska Lincoln
A First Course in Differential Equations Third Edition
March 2, 2015
Springer-Verlag
Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest
To David Russell Logan
Contents
Preface to the Third Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi 1. First-Order Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 First-Order Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Notation and Terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 Growth?Decay Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Geometric Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3 Separable Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.3.1 Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.3.2 Heat Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.3.3 Chemical Reactors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.4 Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.4.1 Integrating Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.4.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 1.4.3 Electrical Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 1.5 One-Dimensional Dynamical Systems . . . . . . . . . . . . . . . . . . . . . . . 54 1.5.1 Autonomous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.5.2 Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 1.5.3 Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2. Second-Order Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1 Classical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1.1 Oscillations and Dissipation . . . . . . . . . . . . . . . . . . . . . . . . . 80 2.2 Equations with Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . 85 2.2.1 The General Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
viii
A First Course in Differential Equations
Third Edition
2.2.2 Real Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 2.2.3 Complex Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.2.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 2.3 Nonhomogeneous Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.3.1 Undetermined Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 2.3.2 Resonance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 2.4 Equations with Variable Coefficients . . . . . . . . . . . . . . . . . . . . . . . . 116 2.4.1 Cauchy?Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 2.4.2 Variation of Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 2.4.3 Reduction of Order* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 2.5 Higher-Order Equations* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2.6 Steady?State Heat Conduction* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3. Laplace Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 3.1 Definition and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 3.2 Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 3.2.1 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 3.3 The Convolution Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3.4 Impulsive Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
4. Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 4.1 Linear Systems vs. Second-Order Equations . . . . . . . . . . . . . . . . . . 178 4.2 Matrices and Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4.2.1 Preliminaries from Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 191 4.2.2 Differential Equations and Equilibria . . . . . . . . . . . . . . . . . . 199 4.3 The Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 4.4 Solving Linear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 4.4.1 Real Unequal Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 4.4.2 Complex Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219 4.4.3 Real, Equal Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4.5 Phase Plane Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 4.6 Nonhomogeneous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
5. Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 5.1 Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248 5.2 Nonlinear Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 5.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5.3.1 The Lotka?Volterra Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5.3.2 Population Ecology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 276 5.3.3 Epidemics; Chemical kinetics . . . . . . . . . . . . . . . . . . . . . . . . 285 5.3.4 Malaria* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295 5.4 Advanced Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
Contents
ix
5.4.1 Periodic Orbits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302 5.5 Bifurcations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
6. Computation of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 6.1 Iteration* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 6.2 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 6.2.1 The Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 6.2.2 The Runge?Kutta Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 6.3 Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
Appendix A. Review and Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 A.1 Review Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 A.2 Supplementary Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339
Appendix B. MATLAB R Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . 349 B.1 Coding Algorithms for Differential Equations . . . . . . . . . . . . . . . . 351 B.2 MATLAB's Built-in ODE Solvers . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 B.3 Symbolic Solutions Using dsolve . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 B.4 Other Routines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 B.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
Preface to the Third Edition
This new edition remains in step with the goals of earlier editions, namely, to offer a concise treatment of basic topics covered in a post-calculus differential equations course. It is written for students in engineering, biosciences, physics, economics, and mathematics. As such, the text is strongly guided by applications in those areas.
The last twenty-five years witnessed dramatic changes in basic calculus courses and in differential equations. One driver of change has been the availability of technology and its role in a standard course, and another is the level of preparation of students with regard to their ability to perform analytical manipulations. Writing a text for such a diverse audience poses a substantial challenge. Some students need only know what a differential equation means and what it implies qualitatively to understand concepts in their areas; others, who plan on taking advanced courses in engineering or the physical sciences where the mathematics is more intense, require ability to perform analytic calculations. This text makes an effort to balance these two issues.
Some outstanding textbooks have been written for this course. But many are calculus-like and voluminous, with extensive graphics, marginal notes, and numerous examples and exercises; they cover many, many more topics than can be discussed in a one-semester course. I have often felt that students become overwhelmed, distracted, and even insecure about skipping material and
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