Elementary Differential Equations and Boundary Value Problems

Description

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. 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