Elementary Differential Equations

Elementary Differential Equations, 11th Edition is written from the viewpoint of the applied mathematician, whose interest in differential equations may sometimes be quite theoretical, sometimes intensely practical, and often somewhere in between. The authors have sought to combine a sound and accurate (but not abstract) exposition of the elementary theory of differential equations with considerable material on methods of solution, analysis, and approximation that have proved useful in a wide variety of applications. While the general structure of the book remains unchanged, some notable changes have been made to improve the clarity and readability of basic material about differential equations and their applications. In addition to expanded explanations, the 11th edition includes new problems, updated figures and examples to help motivate students. The program is primarily intended for undergraduate students of mathematics, science, or engineering, who typically take a course on differential equations during their first or second year of study. The main prerequisite for engaging with the program is a working knowledge of calculus, gained from a normal two] or three] semester course sequence or its equivalent. Some familiarity with matrices will also be helpful in the chapters on systems of differential equations. Cover 1 Title Page 3 Copyright 4 Dedication 5 The Authors 6 Preface 7 Brief Contents 10 Contents 11 CHAPTER 1: Introduction 13 1.1. Some Basic Mathematical Models; Direction Fields 13 Problems 20 1.2. Solutions of Some Differential Equations 21 Problems 27 1.3. Classification of Differential Equations 28 Problems 34 References 35 CHAPTER 2: First-Order Differential Equations 36 2.1. Linear Differential Equations; Method of Integrating Factors 36 Problems 43 2.2. Separable Differential Equations 45 Problems 50 2.3. Modeling with First-Order Differential Equations 51 Problems 59 2.4. Differences Between Linear and Nonlinear Differential Equations 63 Problems 69 2.5 Autonomous Differential Equations and Population Dynamics 70 Problems 79 2.6. Exact Differential Equations and Integrating Factors 82 Problems 87 2.7. Numerical Approximations: Euler’s Method 88 Problems 94 2.8. The Existence and Uniqueness Theorem 95 Problems 102 2.9. First-Order Difference Equations 103 Problems 111 Chapter Review Problems 112 References 113 CHAPTER 3: Second-Order Linear Differential Equations 115 3.1. Homogeneous Differential Equations with Constant Coefficients 115 Problems 121 3.2. Solutions of Linear Homogeneous Equations; the Wronskian 122 Problems 131 3.3. Complex Roots of the Characteristic Equation 132 Problems 137 3.4. Repeated Roots; Reduction of Order 139 Problems 144 3.5. Nonhomogeneous Equations; Method of Undetermined Coefficients 145 Problems 153 3.6. Variation of Parameters 154 Problems 158 3.7. Mechanical and Electrical Vibrations 159 Problems 169 3.8. Forced Periodic Vibrations 171 Problems 179 References 180 CHAPTER 4: Higher-Order Linear Differential Equations 181 4.1. General Theory of nth Order Linear Differential Equations 181 Problems 185 4.2. Homogeneous Differential Equations with Constant Coefficients 186 Problems 192 4.3. The Method of Undetermined Coefficients 193 Problems 196 4.4. The Method of Variation of Parameters 197 Problems 200 References 200 CHAPTER 5: Series Solutions of Second-Order Linear Equations 201 5.1. Review of Power Series 201 Problems 207 5.2. Series Solutions Near an Ordinary Point, Part I 207 Problems 216 5.3. Series Solutions Near an Ordinary Point, Part II 217 Problems 221 5.4. Euler Equations; Regular Singular Points 223 Problems 230 5.5. Series Solutions Near a Regular Singular Point, Part I 231 Problems 235 5.6. Series Solutions Near a Regular Singular Point, Part II 236 Problems 241 5.7. Bessel’s Equation 242 Problems 251 References 252 CHAPTER 6: The Laplace Transform 253 6.1. Definition of the Laplace Transform 253 Problems 259 6.2. Solution of Initial Value Problems 260 Problems 267 6.3. Step Functions 269 Problems 274 6.4. Differential Equations with Discontinuous Forcing Functions 276 Problems 280 6.5. Impulse Functions 282 Problems 285 6.6. The Convolution Integral 287 Problems 291 References 292 CHAPTER 7: Systems of First-Order Linear Equations 293 7.1. Introduction 293 Problems 296 7.2. Matrices 298 Problems 305 7.3. Systems of Linear Algebraic Equations; Linear Independence, Eigenvalues, Eigenvectors 307 Problems 315 7.4. Basic Theory of Systems of First-Order Linear Equations 316 Problems 320 7.5. Homogeneous Linear Systems with Constant Coefficients 321 Problems 330 7.6. Complex-Valued Eigenvalues 331 Problems 339 7.7 Fundamental Matrices 341 Problems 348 7.8. Repeated Eigenvalues 349 Problems 355 7.9. Nonhomogeneous Linear Systems 357 Problems 363 References 365 CHAPTER 8: Numerical Methods 366 8.1. The Euler or Tangent Line Method 366 Problems 373 8.2. Improvements on the Euler Method 375 Problems 378 8.3. The Runge-Kutta Method 379 Problems 382 8.4. Multistep Methods 383 Problems 387 8.5. Systems of First-Order Equations 388 Problems 390 8.6. More on Errors; Stability 390 Problems 398 References 399 CHAPTER 9: Nonlinear Differential Equations and Stability 400 9.1. The Phase Plane: Linear Systems 400 Problems 409 9.2. Autonomous Systems and Stability 410 Problems 418 9.3. Locally Linear Systems 419 Problems 427 9.4. Competing Specie 429 Problems 438 9.5. Predator - Prey Equations 440 Problems 445 9.6. Liapunov’s Second Method 447 Problems 455 9.7. Periodic Solutions and Limit Cycles 456 Problems 464 9.8. Chaos and Strange Attractors: The Lorenz Equations 466 Problems 472 References 473 CHAPTER 10: Partial Differential Equations and Fourier Series 475 10.1. Two-Point Boundary Value Problems 475 Problems 480 10.2. Fourier Series 481 Problems 488 10.3. The Fourier Convergence Theorem 489 Problems 493 10.4. Even and Odd Functions 494 Problems 499 10.5. Separation of Variables; Heat Conduction in a Rod 500 Problems 507 10.6. Other Heat Conduction Problems 508 Problems 514 10.7. The Wave Equation: Vibrations of an Elastic String 516 Problems 524 10.8. Laplace’s Equation 526 Problems 532 A. APPENDIX 534 B. APPENDIX 538 References 540 CHAPTER 11: Boundary Value Problems and Sturm-Liouville Theory 541 11.1. The Occurrence of Two-Point Boundary Value Problems 541 Problems 545 11.2 Sturm-Liouville Boundary Value Problems 547 Problems 555 11.3. Nonhomogeneous Boundary Value Problems 557 Problems 565 11.4. Singular Sturm-Liouville Problems 568 Problems 573 11.5. Further Remarks on the Method of Separation of Variables: A Bessel Series Expansion 574 Problems 576 11.6. Series of Orthogonal Functions: Mean Convergence 578 Problems 583 References 584 Answers to Problems 585 Chapter 1 585 Section 1.1, page 8 585 Section 1.2, page 15 585 Section 1.3, page 22 585 Chapter 2 586 Section 2.1, page 31 586 Section 2.2, page 38 586 Section 2.3, page 47 586 Section 2.4, page 57 587 Section 2.5, page 67 587 Section 2.6, page 75 588 Section 2.7, page 82 588 Section 2.8, page 90 588 Section 2.9, page 99 588 Miscellaneous Problems, page 100 589 Chapter 3 589 Section 3.1, page 109 589 Section 3.2, page 119 589 Section 3.3, page 125 590 Section 3.4, page 132 590 Section 3.5, page 141 591 Section 3.6, page 146 591 Section 3.7, page 157 591 Section 3.8, page 167 592 Chapter 4 592 Section 4.1, page 173 592 Section 4.2, page 180 592 Section 4.3, page 184 593 Section 4.4, page 188 593 Chapter 5 593 Section 5.1, page 195 593 Section 5.2, page 204 594 Section 5.3, page 209 594 Section 5.4, page 218 595 Section 5.5, page 223 595 Section 5.6, page 229 596 Section 5.7, page 239 596 Chapter 6 597 Section 6.1, page 247 597 Section 6.2, page 255 597 Section 6.3, page 262 597 Section 6.4, page 268 598 Section 6.5, page 273 598 Section 6.6, page 279 598 Chapter 7 599 Section 7.1, page 284 599 Section 7.2, page 293 599 Section 7.3, page 303 600 Section 7.4, page 308 600 Section 7.5, page 318 601 Section 7.6, page 327 601 Section 7.7, page 336 602 Section 7.8, page 343 602 Section 7.9, page 351 603 Chapter 8 604 Section 8.1, page 361 604 Section 8.2, page 366 604 Section 8.3, page 370 605 Section 8.4, page 375 605 Section 8.5, page 378 605 Section 8.6, page 386 605 Chapter 9 606 Section 9.1, page 397 606 Section 9.2, page 406 606 Section 9.3, page 415 606 Section 9.4, page 426 607 Section 9.5, page 433 608 Section 9.7, page 452 609 Section 9.8, page 460 609 Chapter 10 609 Section 10.1, page 468 609 Section 10.2, page 476 610 Section 10.3, page 481 610 Section 10.4, page 487 611 Section 10.5, page 495 612 Section 10.6, page 502 612 Section 10.7, page 512 613 Section 10.8, page 520 613 Section 11.1, page 533 614 Section 11.2, page 543 614 Section 11.3, page 553 615 Section 11.4, page 561 616 Section 11.5, page 564 616 Section 11.6, page 571 616 Index 618