Differential Equations with Linear Algebra

Contents......Page 6
Introduction......Page 12
1.1 Motivating problems......Page 22
1.2 Systems of linear equations......Page 27
1.2.1 Row reduction using Maple......Page 34
1.3 Linear combinations......Page 40
1.3.1 Markov chains: an application of matrix-vector multiplication......Page 45
1.3.2 Matrix products using Maple......Page 48
1.4 The span of a set of vectors......Page 52
1.5 Systems of linear equations revisited......Page 58
1.6 Linear independence......Page 68
1.7 Matrix algebra......Page 77
1.7.1 Matrix algebra using Maple......Page 81
1.8 The inverse of a matrix......Page 85
1.8.1 Computer graphics......Page 89
1.8.2 Matrix inverses using Maple......Page 92
1.9 The determinant of a matrix......Page 97
1.9.1 Determinants using Maple......Page 101
1.10 The eigenvalue problem......Page 103
1.10.1 Markov chains, eigenvectors, and Google......Page 112
1.10.2 Using Maple to find eigenvalues and eigenvectors......Page 113
1.11 Generalized vectors......Page 118
1.12 Bases and dimension in vector spaces......Page 127
1.13.1 Computer graphics: geometry and linear algebra at work......Page 134
1.13.2 Bézier curves......Page 138
1.13.3 Discrete dynamical systems......Page 142
2.1 Motivating problems......Page 146
2.2 Definitions, notation, and terminology......Page 148
2.2.1 Plotting slope fields using Maple......Page 154
2.3 Linear first-order differential equations......Page 158
2.4.1 Mixing problems......Page 166
2.4.2 Exponential growth and decay......Page 167
2.4.3 Newton’s law of Cooling......Page 169
2.5.1 Separable equations......Page 173
2.5.2 Exact equations......Page 176
2.6 Euler’s method......Page 181
2.6.1 Implementing Euler’s method in Excel......Page 186
2.7.1 The logistic equation......Page 191
2.7.2 Torricelli’s law......Page 195
2.8.1 Converting certain second-order des to first-order DEs......Page 200
2.8.2 How raindrops fall......Page 201
2.8.3 Riccati’s equation......Page 202
2.8.4 Bernoulli’s equation......Page 203
3.1 Motivating problems......Page 206
3.2 The eigenvalue problem revisited......Page 210
3.3 Homogeneous linear first-order systems......Page 221
3.4 Systems with all real linearly independent eigenvectors......Page 230
3.4.1 Plotting direction fields for systems using Maple......Page 238
3.5 When a matrix lacks two real linearly independent eigenvectors......Page 242
3.6 Nonhomogeneous systems: undetermined coefficients......Page 255
3.7 Nonhomogeneous systems: variation of parameters......Page 264
3.7.1 Applying variation of parameters using Maple......Page 269
3.8.1 Mixing problems......Page 272
3.8.2 Spring-mass systems......Page 274
3.8.3 RLC circuits......Page 277
3.9.1 Diagonalizable matrices and coupled systems......Page 287
3.9.2 Matrix exponential......Page 289
4.1 Motivating equations......Page 292
4.2 Homogeneous equations: distinct real roots......Page 293
4.3.1 Repeated roots......Page 300
4.3.2 Complex roots......Page 302
4.4 Nonhomogeneous equations......Page 307
4.4.1 Undetermined coefficients......Page 308
4.4.2 Variation of parameters......Page 314
4.5 Forced motion: beats and resonance......Page 319
4.6 Higher order linear differential equations......Page 328
4.6.1 Solving characteristic equations using Maple......Page 335
4.7.1 Damped motion......Page 338
4.7.2 Forced oscillations with damping......Page 340
4.7.3 The Cauchy–Euler equation......Page 342
4.7.4 Companion systems and companion matrices......Page 344
5.1 Motivating problems......Page 348
5.2 Laplace transforms: getting started......Page 350
5.3 General properties of the Laplace transform......Page 356
5.4.1 The Heaviside function......Page 366
5.4.2 The Dirac delta function......Page 372
5.4.3 The Heaviside and Dirac functions in Maple......Page 376
5.5 Solving IVPs with the Laplace transform......Page 378
5.6 More on the inverse Laplace transform......Page 390
5.6.1 Laplace transforms and inverse transforms using Maple......Page 394
5.7.1 Laplace transforms of infinite series......Page 397
5.7.2 Laplace transforms of periodic forcing functions......Page 399
5.7.3 Laplace transforms of systems......Page 403
6.1 Motivating problems......Page 406
6.2 Graphical behavior of solutions for 2 × 2 nonlinear systems......Page 410
6.2.1 Plotting direction fields of nonlinear systems using Maple......Page 416
6.3 Linear approximations of nonlinear systems......Page 419
6.4 Euler’s method for nonlinear systems......Page 428
6.4.1 Implementing Euler’s method for systems in Excel......Page 432
6.5.1 The damped pendulum......Page 436
6.5.2 Competitive species......Page 437
7.1 Motivating problems......Page 440
7.2 Beyond Euler’s method......Page 442
7.2.1 Heun’s method......Page 443
7.2.2 Modified Euler’s method......Page 446
7.3 Higher order methods......Page 449
7.3.1 Taylor methods......Page 450
7.3.2 Runge–Kutta methods......Page 453
7.4 Methods for systems and higher order equations......Page 458
7.4.1 Euler’s method for systems......Page 459
7.4.2 Heun’s method for systems......Page 461
7.4.3 Runge–Kutta method for systems......Page 462
7.4.4 Methods for higher order IVPs......Page 464
7.5.1 Predator–Prey equations......Page 468
7.5.3 The damped pendulum......Page 469
8.1 Motivating problems......Page 472
8.2 A review of Taylor and power series......Page 474
8.3 Power series solutions of linear equations......Page 482
8.4 Legendre’s equation......Page 490
8.5.1 The Hermite equation......Page 496
8.5.2 The Laguerre equation......Page 499
8.5.3 The Bessel equation......Page 501
8.6 The method of Frobenius......Page 504
8.7.2 The Gamma function......Page 510
Appendix A: Review of integration techniques......Page 512
Appendix B: Complex numbers......Page 522
Appendix C: Roots of polynomials......Page 528
Appendix D: Linear transformations......Page 532
Appendix E: Solutions to selected exercises......Page 542
C......Page 568
I......Page 569
N......Page 570
S......Page 571
Z......Page 572