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: 9
Publisher: Wiley
Year: 2008
Language: English
Pages: 818
Tags: Математика;Дифференциальные уравнения;
Cover Page......Page 1
Title Page......Page 7
Copyright Page......Page 8
Dedication......Page 9
The Authors......Page 10
PREFACE......Page 11
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 69
2.3 Modeling with First Order Equations......Page 72
PROBLEMS......Page 81
Theorem 2.4.1......Page 90
Theorem 2.4.2......Page 92
PROBLEMS......Page 97
2.5 Autonomous Equations and Population Dynamics......Page 100
PROBLEMS......Page 110
2.6 Exact Equations and Integrating Factors......Page 116
Theorem 2.6.1......Page 117
PROBLEMS......Page 121
2.7 Numerical Approximations: Euler’s Method......Page 123
PROBLEMS......Page 131
2.8 The Existence and Uniqueness Theorem......Page 133
Theorem 2.8.1......Page 134
PROBLEMS......Page 140
2.9 First Order Difference Equations......Page 143
PROBLEMS......Page 152
PROBLEMS......Page 154
REFERENCES......Page 157
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
PROBLEMS......Page 177
3.3 Complex Roots of the Characteristic Equation......Page 179
PROBLEMS......Page 185
3.4 Repeated Roots; Reduction of Order......Page 188
PROBLEMS......Page 193
3.5 Nonhomogeneous Equations; Method of Undetermined Coefficients......Page 196
PROBLEMS......Page 205
3.6 Variation of Parameters......Page 207
Theorem 3.6.1......Page 210
PROBLEMS......Page 211
3.7 Mechanical and Electrical Vibrations......Page 213
PROBLEMS......Page 224
3.8 Forced Vibrations......Page 228
PROBLEMS......Page 237
REFERENCES......Page 239
4.1 General Theory of nth Order Linear Equations......Page 241
PROBLEMS......Page 246
4.2 Homogeneous Equations with Constant Coefficients......Page 248
PROBLEMS......Page 253
4.3 The Method of Undetermined Coefficients......Page 256
PROBLEMS......Page 259
4.4 The Method of Variation of Parameters......Page 261
PROBLEMS......Page 264
5.1 Review of Power Series......Page 265
PROBLEMS......Page 271
5.2 Series Solutions Near an Ordinary Point, Part I......Page 272
PROBLEMS......Page 281
5.3 Series Solutions Near an Ordinary Point, Part II......Page 283
Theorem 5.3.1......Page 284
PROBLEMS......Page 287
5.4 Euler Equations; Regular Singular Points......Page 290
PROBLEMS......Page 298
5.5 Series Solutions Near a Regular Singular Point, Part I......Page 300
PROBLEMS......Page 304
5.6 Series Solutions Near a Regular Singular Point, Part II......Page 306
Theorem 5.6.1......Page 311
PROBLEMS......Page 312
5.7 Bessel’s Equation......Page 314
PROBLEMS......Page 323
REFERENCES......Page 326
6.1 Definition of the Laplace Transform......Page 327
Theorem 6.1.1......Page 329
Theorem 6.1.2......Page 330
PROBLEMS......Page 333
6.2 Solution of Initial Value Problems......Page 334
Theorem 6.2.1......Page 335
Corollary 6.2.2......Page 336
PROBLEMS......Page 342
6.3 Step Functions......Page 345
PROBLEMS......Page 350
6.4 Differential Equations with Discontinuous Forcing Functions......Page 353
PROBLEMS......Page 358
6.5 Impulse Functions......Page 361
PROBLEMS......Page 365
Theorem 6.6.1......Page 367
PROBLEMS......Page 372
REFERENCES......Page 375
7.1 Introduction......Page 377
Theorem 7.1.1......Page 380
PROBLEMS......Page 381
7.2 Review of Matrices......Page 386
PROBLEMS......Page 393
7.3 Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors......Page 395
PROBLEMS......Page 405
7.4 Basic Theory of Systems of First Order Linear Equations......Page 407
Theorem 7.4.1......Page 408
Theorem 7.4.3......Page 409
Theorem 7.4.4......Page 410
PROBLEMS......Page 411
7.5 Homogeneous Linear Systems with Constant Coefficients......Page 412
PROBLEMS......Page 420
7.6 Complex Eigenvalues......Page 423
PROBLEMS......Page 431
7.7 Fundamental Matrices......Page 435
PROBLEMS......Page 442
7.8 Repeated Eigenvalues......Page 444
PROBLEMS......Page 450
7.9 Nonhomogeneous Linear Systems......Page 454
PROBLEMS......Page 461
REFERENCES......Page 463
8.1 The Euler or Tangent Line Method......Page 465
PROBLEMS......Page 473
8.2 Improvements on the Euler Method......Page 476
PROBLEMS......Page 480
8.3 The Runge–Kutta Method......Page 481
PROBLEMS......Page 485
8.4 Multistep Methods......Page 486
PROBLEMS......Page 491
8.5 More on Errors; Stability......Page 492
PROBLEMS......Page 501
8.6 Systems of First Order Equations......Page 502
PROBLEMS......Page 505
REFERENCES......Page 506
9.1 The Phase Plane: Linear Systems......Page 507
PROBLEMS......Page 516
9.2 Autonomous Systems and Stability......Page 519
PROBLEMS......Page 528
Theorem 9.3.1......Page 530
Theorem 9.3.2
......Page 534
PROBLEMS......Page 538
9.4 Competing Species......Page 542
PROBLEMS......Page 552
9.5 Predator–Prey Equations......Page 555
PROBLEMS......Page 562
9.6 Liapunov’s Second Method......Page 565
Theorem 9.6.2......Page 569
Theorem 9.6.3......Page 571
Theorem 9.6.4......Page 572
PROBLEMS......Page 573
9.7 Periodic Solutions and Limit Cycles......Page 576
Theorem 9.7.2......Page 579
Theorem 9.7.3......Page 580
PROBLEMS......Page 585
9.8 Chaos and Strange Attractors: The Lorenz Equations......Page 588
PROBLEMS......Page 595
REFERENCES......Page 597
10.1 Two-Point Boundary Value Problems......Page 599
PROBLEMS......Page 605
10.2 Fourier Series......Page 606
PROBLEMS......Page 614
10.3 The Fourier Convergence Theorem......Page 617
Theorem 10.3.1......Page 618
PROBLEMS......Page 622
10.4 Even and Odd Functions......Page 624
PROBLEMS......Page 630
10.5 Separation of Variables; Heat Conduction in a Rod......Page 633
PROBLEMS......Page 640
10.6 Other Heat Conduction Problems......Page 642
PROBLEMS......Page 649
10.7 The Wave Equation: Vibrations of an Elastic String......Page 653
PROBLEMS......Page 662
10.8 Laplace’s Equation......Page 668
PROBLEMS......Page 675
REFERENCES......Page 685
APPENDIX A
......Page 679
APPENDIX B......Page 683
11.1 The Occurrence of Two-Point Boundary Value Problems......Page 687
PROBLEMS......Page 692
11.2 Sturm–Liouville Boundary Value Problems......Page 695
Theorem 11.2.1......Page 698
Theorem 11.2.3......Page 699
Theorem 11.2.4......Page 702
PROBLEMS......Page 705
11.3 Nonhomogeneous Boundary Value Problems......Page 709
Theorem 11.3.2......Page 712
PROBLEMS......Page 718
11.4 Singular Sturm–Liouville Problems......Page 724
PROBLEMS......Page 729
11.5 Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion......Page 731
PROBLEMS......Page 735
11.6 Series of Orthogonal Functions: Mean Convergence......Page 738
Theorem 11.6.1......Page 742
PROBLEMS......Page 744
REFERENCES......Page 746
Answers to Problems......Page 749
Index......Page 808
