Differential Equations with Boundary-Value Problems

Description

DIFFERENTIAL EQUATIONS WITH BOUNDARY-VALUE PROBLEMS, 9th Edition, strikes a balance between the analytical, qualitative, and quantitative approaches to the study of Differential Equations. This proven text speaks to students of varied majors through a wealth of pedagogical aids, including an abundance of examples, explanations, "Remarks" boxes, and definitions. Written in a straightforward, readable, and helpful style, the book provides a thorough overview of the topics typically taught in a first course in Differential Equations as well as an introduction to boundary-value problems and partial Differential Equations. Cover......Page 1 Contents......Page 4 Preface......Page 8 Ch 1: Introduction to Differential Equations......Page 13 1.1 Definitions and Terminology......Page 14 1.2 Initial-Value Problems......Page 26 1.3 Differential Equations as Mathematical Models......Page 33 Chapter 1 In Review......Page 45 Ch 2: First-Order Differential Equations......Page 47 2.1 Solution Curves Without a Solution......Page 48 2.2 Separable Equations......Page 58 2.3 Linear Equations......Page 66 2.4 Exact Equations......Page 75 2.5 Solutions by Substitutions......Page 83 2.6 A Numerical Method......Page 87 Chapter 2 In Review......Page 92 Ch 3: Modeling with First-Order Differential Equations......Page 95 3.1 Linear Models......Page 96 3.2 Nonlinear Models......Page 107 3.3 Modeling with Systems of First-Order DEs......Page 118 Chapter 3 In Review......Page 125 Ch 4: Higher-Order Differential Equations......Page 129 4.1 Preliminary Theory-Linear Equations......Page 130 4.2 Reduction of Order......Page 143 4.3 Homogeneous Linear Equations with Constant Coefficients......Page 146 4.4 Undetermined Coefficients-Superposition Approach......Page 153 4.5 Undetermined Coefficients-Annihilator Approach......Page 163 4.6 Variation of Parameters......Page 170 4.7 Cauchy-Euler Equations......Page 177 4.8 Green's Functions......Page 184 4.9 Solving Systems of Linear DEs by Elimination......Page 194 4.10 Nonlinear Differential Equations......Page 199 Chapter 4 In Review......Page 204 Ch 5: Modeling with Higher-Order Differential Equations......Page 207 5.1 Linear Models: Initial-Value Problems......Page 208 5.2 Linear Models: Boundary-Value Problems......Page 224 5.3 Nonlinear Models......Page 233 Chapter 5 In Review......Page 243 Ch 6: Series Solutions of Linear Equations......Page 247 6.1 Review of Power Series......Page 248 6.2 Solutions about Ordinary Points......Page 254 6.3 Solutions about Singular Points......Page 263 6.4 Special Functions......Page 273 Chapter 6 In Review......Page 287 Ch 7: The Laplace Transform......Page 289 7.1 Definition of the Laplace Transform......Page 290 7.2 Inverse Transforms and Transforms of Derivatives......Page 297 7.3 Operational Properties I......Page 305 7.4 Operational Properties II......Page 317 7.5 The Dirac Delta Function......Page 329 7.6 Systems of Linear Differential Equations......Page 333 Chapter 7 In Review......Page 338 Ch 8: Systems of Linear First-Order Differential Equations......Page 343 8.1 Preliminary Theory-Linear Systems......Page 344 8.2 Homogeneous Linear Systems......Page 351 8.3 Nonhomogeneous Linear Systems......Page 366 8.4 Matrix Exponential......Page 373 Chapter 8 In Review......Page 377 Ch 9: Numerical Solutions of Ordinary Differential Equations......Page 379 9.1 Euler Methods and Error Analysis......Page 380 9.2 Runge-Kutta Methods......Page 385 9.3 Multistep Methods......Page 389 9.4 Higher-Order Equations and Systems......Page 392 9.5 Second-Order Boundary-Value Problems......Page 396 Chapter 9 In Review......Page 400 Ch 10: Systems of Nonlinear First-Order Differential Equations......Page 401 10.1 Autonomous Systems......Page 402 10.2 Stability of Linear Systems......Page 408 10.3 Linearization and Local Stability......Page 416 10.4 Autonomous Systems as Mathematical Models......Page 425 Chapter 10 In Review......Page 433 Ch 11: Fourier Series......Page 435 11.1 Orthogonal Functions......Page 436 11.2 Fourier Series......Page 442 11.3 Fourier Cosine and Sine Series......Page 447 11.4 Sturm-Liouville Problem......Page 455 11.5 Bessel and Legendre Series......Page 462 Chapter 11 In Review......Page 469 Ch 12: Boundary-Value Problems in Rectangular Coordinates......Page 471 12.1 Separable Partial Differential Equations......Page 472 12.2 Classical PDEs and Boundary-Value Problems......Page 476 12.3 Heat Equation......Page 482 12.4 Wave Equation......Page 484 12.5 Laplace's Equation......Page 490 12.6 Nonhomogeneous Boundary-Value Problems......Page 495 12.7 Orthogonal Series Expansions......Page 502 12.8 Higher-Dimensional Problems......Page 507 Chapter 12 In Review......Page 510 Ch 13: Boundary-Value Problems in Other Coordinate Systems......Page 513 13.1 Polar Coordinates......Page 514 13.2 Polar and Cylindrical Coordinates......Page 519 13.3 Spherical Coordinates......Page 526 Chapter 13 In Review......Page 528 Ch 14: Integral Transforms......Page 531 14.1 Error Function......Page 532 14.2 Laplace Transform......Page 533 14.3 Fourier Integral......Page 541 14.4 Fourier Transforms......Page 547 Chapter 14 In Review......Page 553 Ch 15: Numerical Solutions of Partial Differential Equations......Page 555 15.1 Laplace's Equation......Page 556 15.2 Heat Equation......Page 561 15.3 Wave Equation......Page 566 Chapter 15 In Review......Page 570 Appendices......Page 571 Appendix A: Integral-Defined Functions......Page 572 Appendix B: Matrices......Page 580 Appendix C: Laplace Transforms......Page 598 Answers for Selected Odd-Numbered Problems......Page 602 Index......Page 632