Written from the perspective of the applied mathematician, the latest edition of this bestselling book focuses on the theory and practical applications of Differential Equations to engineering and the sciences. Emphasis is placed on the methods of solution, analysis, and approximation. Use of technology, illustrations, and problem sets help readers develop an intuitive understanding of the material. Historical footnotes trace the development of the discipline and identify outstanding individual contributions. This book builds the foundation for anyone who needs to learn differential equations and then progress to more advanced studies.
Author(s): William E. Boyce, Richard C. DiPrima
Edition: 10
Publisher: Wiley
Year: 2012
Language: English
Pages: 832
Tags: Математика;Дифференциальные уравнения;
Cover......Page 1
Title Page......Page 7
Copyright Page......Page 8
The Authors
......Page 10
Preface......Page 11
WileyPLUS......Page 15
Acknowledgments......Page 17
Contents......Page 19
1.1: Some Basic Mathematical Models; Direction Fields......Page 23
Problems......Page 29
1.2: Solutions of Some Differential Equations......Page 32
Problems......Page 37
1.3: Classification of Differential Equations......Page 41
Problems......Page 46
1.4: Historical Remarks......Page 48
References......Page 51
2.1: Linear Equations; Method of Integrating Factors......Page 53
Problems......Page 61
2.2: Separable Equations......Page 64
Problems......Page 70
2.3: Modeling with First Order Equations......Page 73
Problems......Page 82
2.4: Differences Between Linear and Nonlinear Equations......Page 90
Theorem 2.4.1......Page 91
Theorem 2.4.2......Page 92
Problems......Page 98
2.5: Autonomous Equations and Population Dynamics......Page 100
Problems......Page 110
2.6: Exact Equations and Integrating Factors......Page 117
Theorem 2.6.1......Page 118
Problems......Page 123
2.7: Numerical Approximations: Euler's Method......Page 124
Problems......Page 132
2.8: The Existence and Uniqueness Theorem......Page 134
Theorem 2.8.1......Page 135
Problems......Page 142
2.9: First Order Difference Equations......Page 144
Problems......Page 153
Problems......Page 155
References......Page 158
3.1: Homogeneous Equations with Constant Coefficients......Page 159
Problems......Page 166
3.2: Solutions of Linear Homogeneous Equations; the Wronskian......Page 167
Theorem 3.2.1......Page 168
Theorem 3.2.2......Page 169
Theorem 3.2.4......Page 171
Theorem 3.2.5......Page 173
Theorem 3.2.6......Page 175
Theorem 3.2.7......Page 176
Problems......Page 177
3.3: Complex Roots of the Characteristic Equation......Page 180
Problems......Page 186
3.4: Repeated Roots; Reduction of Order......Page 189
Problems......Page 194
3.5: Nonhomogeneous Equations; Method of Undetermined Coefficients......Page 197
Theorem 3.5.2......Page 198
Problems......Page 206
3.6: Variation of Parameters......Page 208
Theorem 3.6.1......Page 211
Problems......Page 212
3.7: Mechanical and Electrical Vibrations......Page 214
Problems......Page 225
3.8: Forced Vibrations......Page 229
Problems......Page 239
References......Page 241
4.1: General Theory of nth Order Linear Equations......Page 243
Theorem 4.1.1......Page 244
Theorem 4.1.2......Page 245
Theorem 4.1.3......Page 247
Problems......Page 248
4.2: Homogeneous Equations with Constant Coefficients......Page 250
Problems......Page 255
4.3: The Method of Undetermined Coefficients......Page 258
Problems......Page 261
4.4: The Method of Variation of Parameters......Page 263
Problems......Page 266
References......Page 267
5.1: Review of Power Series......Page 269
Problems......Page 275
5.2: Series Solutions Near an Ordinary Point, Part I......Page 276
Problems......Page 285
5.3: Series Solutions Near an Ordinary Point, Part II......Page 287
Theorem 5.3.1......Page 288
Problems......Page 291
5.4: Euler Equations; Regular Singular Points......Page 294
Problems......Page 302
5.5: Series Solutions Near a Regular Singular Point, Part I......Page 304
Problems......Page 308
5.6: Series Solutions Near a Regular Singular Point, Part II......Page 310
Theorem 5.6.1......Page 315
Problems......Page 316
5.7: Bessel's Equation......Page 318
Problems......Page 327
References......Page 330
6.1: Definition of the Laplace Transform......Page 331
Theorem 6.1.1......Page 333
Theorem 6.1.2......Page 334
Problems......Page 337
Theorem 6.2.1......Page 339
Corollary 6.2.2......Page 340
Problems......Page 346
6.3: Step Functions......Page 349
Theorem 6.3.1......Page 352
Theorem 6.3.2......Page 354
Problems......Page 355
6.4: Differential Equations with Discontinuous Forcing Functions......Page 358
Problems......Page 362
6.5: Impulse Functions......Page 365
Problems......Page 370
Theorem 6.6.1......Page 372
Problems......Page 376
References......Page 380
7.1: Introduction......Page 381
Theorem 7.1.1......Page 384
Problems......Page 385
7.2: Review of Matrices......Page 390
Problems......Page 398
7.3: Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors......Page 400
Problems......Page 410
7.4: Basic Theory of Systems of First Order Linear Equations......Page 412
Theorem 7.4.1......Page 413
Theorem 7.4.2......Page 414
Theorem 7.4.4......Page 415
Problems......Page 416
7.5: Homogeneous Linear Systems with Constant Coefficients......Page 418
Problems......Page 427
7.6: Complex Eigenvalues......Page 430
Problems......Page 439
7.7: Fundamental Matrices......Page 443
Problems......Page 449
7.8: Repeated Eigenvalues......Page 451
Problems......Page 458
7.9: Nonhomogeneous Linear Systems......Page 462
Problems......Page 469
References......Page 471
8.1: The Euler or Tangent Line Method......Page 473
Problems......Page 482
8.2: Improvements on the Euler Method......Page 484
Problems......Page 488
8.3: The Runge–Kutta Method......Page 490
Problems......Page 493
8.4: Multistep Methods......Page 494
8.5: Systems of First Order Equations......Page 500
Problems......Page 503
8.6: More on Errors; Stability......Page 504
Problems......Page 513
References......Page 515
9.1: The Phase Plane: Linear Systems......Page 517
Problems......Page 527
9.2: Autonomous Systems and Stability......Page 530
Problems......Page 539
Theorem 9.3.1......Page 541
Theorem 9.3.2......Page 545
Problems......Page 549
9.4: Competing Species......Page 553
Problems......Page 563
9.5: Predator–Prey Equations......Page 566
Problems......Page 573
9.6: Liapunov's Second Method......Page 576
Theorem 9.6.2......Page 580
Theorem 9.6.3......Page 582
Theorem 9.6.4......Page 583
Problems......Page 584
9.7: Periodic Solutions and Limit Cycles......Page 587
Theorem 9.7.2......Page 590
Theorem 9.7.3......Page 591
Problems......Page 596
9.8: Chaos and Strange Attractors: The Lorenz Equations......Page 599
Problems......Page 606
References......Page 609
10.1: Two-Point Boundary Value Problems......Page 611
Problems......Page 617
10.2: Fourier Series......Page 618
Problems......Page 627
10.3: The Fourier Convergence Theorem......Page 629
Theorem 10.3.1......Page 630
Problems......Page 634
10.4: Even and Odd Functions......Page 636
Problems......Page 642
10.5: Separation of Variables; Heat Conduction in a Rod......Page 645
Problems......Page 652
10.6: Other Heat Conduction Problems......Page 654
Problems......Page 661
10.7: The Wave Equation: Vibrations of an Elastic String......Page 665
Problems......Page 674
10.8: Laplace's Equation......Page 680
Problems......Page 687
Appendix A: Derivation of the Heat Conduction Equation......Page 691
Appendix B: Derivation of the Wave Equation......Page 695
References......Page 697
11.1: The Occurrence of Two-Point Boundary Value Problems......Page 699
Problems......Page 704
11.2: Sturm–Liouville Boundary Value Problems......Page 707
Theorem 11.2.1......Page 710
Theorem 11.2.3......Page 711
Theorem 11.2.4......Page 714
Problems......Page 717
11.3: Nonhomogeneous Boundary Value Problems......Page 721
Theorem 11.3.2......Page 724
Problems......Page 730
11.4: Singular Sturm–Liouville Problems......Page 736
Problems......Page 741
11.5: Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion......Page 743
Problems......Page 747
11.6: Series of Orthogonal Functions: Mean Convergence......Page 750
Theorem 11.6.1......Page 754
Problems......Page 756
References......Page 758
Answers to Problems......Page 761
Index......Page 821
