Differential Equations with Linear Algebra

Linearity plays a critical role in the study of elementary differential equations; linear differential equations, especially systems thereof, demonstrate a fundamental application of linear algebra. In Differential Equations with Linear Algebra, we explore this interplay between linear algebra and differential equations and examine introductory and important ideas in each, usually through the lens of important problems that involve differential equations. Written at a sophomore level, the text is accessible to students who have completed multivariable calculus. With a systems-first approach, the book is appropriate for courses for majors in mathematics, science, and engineering that study systems of differential equations.

Because of its emphasis on linearity, the text opens with a full chapter devoted to essential ideas in linear algebra. Motivated by future problems in systems of differential equations, the chapter on linear algebra introduces such key ideas as systems of algebraic equations, linear combinations, the eigenvalue problem, and bases and dimension of vector spaces. This chapter enables students to quickly learn enough linear algebra to appreciate the structure of solutions to linear differential equations and systems thereof in subsequent study and to apply these ideas regularly.

The book offers an example-driven approach, beginning each chapter with one or two motivating problems that are applied in nature. The following chapter develops the mathematics necessary to solve these problems and explores related topics further. Even in more theoretical developments, we use an example-first style to build intuition and understanding before stating or proving general results. Over 100 figures provide visual demonstration of key ideas; the use of the computer algebra system Maple and Microsoft Excel are presented in detail throughout to provide further perspective and support students' use of technology in solving problems. Each chapter closes with several substantial projects for further study, many of which are based in applications.

Errata sheet available at: www.oup.com/us/companion.websites/9780195385861/pdf/errata.pdf

Matt Boelkins is Associate Professor of Mathematics at Grand Valley State University.
Merle C. Potter is Professor Emeritus of Engineering at Michigan State University and was the first recipient of the Teacher-Scholar award. He has authored or coauthored twenty-four textbooks and exam review books.
Jack Goldberg is Professor Emeritus of Mathematics at the University of Michigan. He has published several textbooks and numerous research papers.

Introduction	p. xi
Essentials of linear algebra	p. 3
Motivating problems	p. 3
Systems of linear equations	p. 8
Row reduction using Maple	p. 15
Linear combinations	p. 21
Markov chains: an application of matrix-vector multiplication	p. 26
Matrix products using Maple	p. 29
The span of a set of vectors	p. 33
Systems of linear equations revisited	p. 39
Linear independence	p. 49
Matrix algebra	p. 58
Matrix algebra using Maple	p. 62
The inverse of a matrix	p. 66
Computer graphics	p. 70
Matrix inverses using Maple	p. 73
The determinant of a matrix	p. 78
Determinants using Maple	p. 82
The eigenvalue problem	p. 84
Markov chains, eigenvectors, and Google	p. 93
Using Maple to find eigenvalues and eigenvectors	p. 94
Generalized vectors	p. 99
Bases and dimension in vector spaces	p. 108
For further study	p. 115
Computer graphics: geometry and linear algebra at work	p. 115
Bézier curves	p. 119
Discrete dynamical systems	p. 123
First-order differential equations	p. 127
Motivating problems	p. 127
Definitions, notation, and terminology	p. 129
Plotting slope fields using Maple	p. 135
Linear first-order differential equations	p. 139
Applications of linear first-order differential equations	p. 147
Mixing problems	p. 147
Exponential growth and decay	p. 148
Newton's law of Cooling	p. 150
Nonlinear first-order differential equations	p. 154
Separable equations	p. 154
Exact equations	p. 157
Euler's method	p. 162
Implementing Euler's method in Excel	p. 167
Applications of nonlinear first-order differential equations	p. 172
The logistic equation	p. 172
Torricelli's law	p. 176
For further study	p. 181
Converting certain second-order des to first-order DEs	p. 181
How raindrops fall	p. 182
Riccati's equation	p. 183
Bernoulli's equation	p. 184
Linear systems of differential equations	p. 187
Motivating problems	p. 187
The eigenvalue problem revisited	p. 191
Homogeneous linear first-order systems	p. 202
Systems with all real linearly independent eigenvectors	p. 211
Plotting direction fields for systems using Maple	p. 219
When a matrix lacks two real linearly independent eigenvectors	p. 223
Nonhomogeneous systems: undetermined coefficients	p. 236
Nonhomogeneous systems: variation of parameters	p. 245
Applying variation of parameters using Maple	p. 250
Applications of linear systems	p. 253
Mixing problems	p. 253
Spring-mass systems	p. 255
RLC circuits	p. 258
For further study	p. 268
Diagonalizable matrices and coupled systems	p. 268
Matrix exponential	p. 270
Higher order differential equations	p. 273
Motivating equations	p. 273
Homogeneous equations: distinct real roots	p. 274
Homogeneous equations: repeated and complex roots	p. 281
Repeated roots	p. 281
Complex roots	p. 283
Nonhomogeneous equations	p. 288
Undetermined coefficients	p. 289
Variation of parameters	p. 295
Forced motion: beats and resonance	p. 300
Higher order linear differential equations	p. 309
Solving characteristic equations using Maple	p. 316
For further study	p. 319
Damped motion	p. 319
Forced oscillations with damping	p. 321
The Cauchy-Euler equation	p. 323
Companion systems and companion matrices	p. 325
Laplace transforms	p. 329
Motivating problems	p. 329
Laplace transforms: getting started	p. 331
General properties of the Laplace transform	p. 337
Piecewise continuous functions	p. 347
The Heaviside functions	p. 347
The Dirac delta function	p. 353
The Heaviside and Dirac functions in Maple	p. 357
Solving IVPs with the Laplace transform	p. 359
More on the inverse Laplace transform	p. 371
Laplace transforms and inverse transforms using Maple	p. 375
For further study	p. 378
Laplace transforms of infinite series	p. 378
Laplace transforms of periodic forcing functions	p. 380
Laplace transforms of systems	p. 384
Nonlinear systems of differential equations	p. 387
Motivating problems	p. 387
Graphical behavior of solutions for 2 × 2 nonlinear systems	p. 391
Plotting direction fields of nonlinear systems using Maple	p. 397
Linear approximations of nonlinear systems	p. 400
Euler's method for nonlinear systems	p. 409
Implementing Euler's method for systems in Excel	p. 413
For further study	p. 417
The damped pendulum	p. 417
Competitive species	p. 418
Numerical methods for differential equations	p. 421
Motivating problems	p. 421
Beyond Euler's method	p. 423
Heun's method	p. 424
Modified Euler's method	p. 427
Higher order methods	p. 430
Taylor methods	p. 431
Runge-Kutta methods	p. 434
Methods for systems and higher order equations	p. 439
Euler's methods for systems	p. 440
Heun's method for systems	p. 442
Runge-Kutta method for systems	p. 443
Methods for higher order IVPs	p. 445
For further study	p. 449
Predator-Prey equations	p. 449
Competitive species	p. 450
The damped pendulum	p. 450
Series solutions for differential equations	p. 453
Motivating problems	p. 453
A review of Taylor and power series	p. 455
Power series solutions of linear equations	p. 463
Legendre's equation	p. 471
Three important examples	p. 477
The Hermite equation	p. 477
The Laguerre equation	p. 480
The Bessel equation	p. 482
The method of Frobenius	p. 485
For further study	p. 491
Taylor series for first-order differential equations	p. 491
The Gamma function	p. 491
Review of integration techniques	p. 493
Complex numbers	p. 503
Roots of polynomials	p. 509
Linear transformations	p. 513
Solutions to selected exercises	p. 523
Index	p. 549
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Differential Equations with Linear Algebra

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