Differential Equations with Linear Algebra

Matthew R. Boelkins, J. L. Goldberg, and Merle C. Potter


Oxford University Press | English | 2009-11-05 | ISBN: 0195385861 | 576 pages | PDF | 3,3 MB


Contents

Introduction xi

1 Essentials of linear algebra 3
1.1 Motivating problems 3
1.2 Systems of linear equations 8
1.3 Linear combinations 21
1.4 The span of a set of vectors 33
1.5 Systems of linear equations revisited 39
1.6 Linear independence 49
1.7 Matrix algebra 58
1.8 The inverse of a matrix 66
1.9 The determinant of a matrix 78
1.10 The eigenvalue problem 84
1.11 Generalized vectors 99
1.12 Bases and dimension in vector spaces 108
1.13 For further study 115

2 First-order differential equations 127
2.1 Motivating problems 127
2.2 Definitions, notation, and terminology 129
2.3 Linear first-order differential equations 139
2.4 Applications of linear first-order differential equations 147
2.5 Nonlinear first-order differential equations 154
2.6 Euler’s method 162
2.7 Applications of nonlinear first-order differential equations 172
2.8 For further study 181

3 Linear systems of differential equations 187
3.1 Motivating problems 187
3.2 The eigenvalue problem revisited 191
3.3 Homogeneous linear first-order systems 202
3.4 Systems with all real linearly independent eigenvectors 211
3.5 When a matrix lacks two real linearly independent eigenvectors 223
3.6 Nonhomogeneous systems: undetermined coefficients 236
3.7 Nonhomogeneous systems: variation of parameters 245
3.8 Applications of linear systems 253
3.9 For further study 268

4 Higher order differential equations 273
4.1 Motivating equations 273
4.2 Homogeneous equations: distinct real roots 274
4.3 Homogeneous equations: repeated and complex roots 281
4.4 Nonhomogeneous equations 288
4.5 Forced motion: beats and resonance 300
4.6 Higher order linear differential equations 309
4.7 For further study 319

5 Laplace transforms 329
5.1 Motivating problems 329
5.2 Laplace transforms: getting started 331
5.3 General properties of the Laplace transform 337
5.4 Piecewise continuous functions 347
5.5 Solving IVPs with the Laplace transform 359
5.6 More on the inverse Laplace transform 371
5.6.1 Laplace transforms and inverse transforms using Maple 375
5.7 For further study 378

6 Nonlinear systems of differential equations 387
6.1 Motivating problems 387
6.2 Graphical behavior of solutions for 2×2 nonlinear systems 391
6.2.1 Plotting direction fields of nonlinear systems using Maple 397
6.3 Linear approximations of nonlinear systems 400
6.4 Euler’s method for nonlinear systems 409
6.5 For further study 417

7 Numerical methods for differential equations 421
7.1 Motivating problems 421
7.2 Beyond Euler’s method 423
7.3 Higher order methods 430
7.4 Methods for systems and higher order equations 439
7.5 For further study 449

8 Series solutions for differential equations 453
8.1 Motivating problems 453
8.2 A review of Taylor and power series 455
8.3 Power series solutions of linear equations 463
8.4 Legendre’s equation 471
8.5 Three important examples 477
8.6 The method of Frobenius 485
8.7 For further study 491

Appendix A Review of integration techniques 493
Appendix B Complex numbers 503
Appendix C Roots of polynomials 509
Appendix D Linear transformations 513
Appendix E Solutions to selected exercises 523
Index 549