Ordinary Differential Equations

Description

Front Cover......Page 1 Table of Contents......Page 6 Preface (with Important Information for the Reader)......Page 14 Part I: The Basics......Page 16 Chapter 1: The Starting Point: Basic Concepts and Terminology......Page 18 Chapter 2: Integration and Differential Equations......Page 36 Part II: First-Order Equations......Page 52 Chapter 3: Some Basics about First-Order Equations......Page 54 Chapter 4: Separable First-Order Equations......Page 82 Chapter 5: Linear First-Order Equations......Page 110 Chapter 6: Simplifying Through Substitution......Page 122 Chapter 7: The Exact Form and General Integrating Factors......Page 134 Chapter 8: Slope Fields: Graphing Solutions Without the Solutions......Page 160 Chapter 9: Euler's Numerical Method......Page 192 Chapter 10: The Art and Science of Modeling with First-Order Equations......Page 212 Part III: Second- and Higher-Order Equations......Page 240 Chapter 11: Higher-Order Equations: Extending First-Order Concepts......Page 242 Chapter 12: Higher-Order Linear Equations and the Reduction of Order Method......Page 260 Chapter 13: General Solutions to Homogeneous Linear Differential Equations......Page 276 Chapter 14: Verifying the Big Theorems and an Introduction to Differential Operators......Page 296 Chapter 15: Second-Order Homogeneous Linear Equations with Constant Coefficients......Page 314 Chapter 16: Springs: Part I......Page 334 Chapter 17: Arbitrary Homogeneous Linear Equations with Constant Coefficients......Page 350 Chapter 18: Euler Equations......Page 370 Chapter 19: Nonhomogeneous Equations in General......Page 384 Chapter 20: Method of Undetermined Coefficients (aka: Method of Educated Guess)......Page 396 Chapter 21: Springs: Part II......Page 416 Chapter 22: Variation of Parameters (A Better Reduction of Order Method)......Page 432 Part IV: The Laplace Transform......Page 448 Chapter 23: The Laplace Transform (Intro)......Page 450 Chapter 24: Differentiation and the Laplace Transform......Page 480 Chapter 25: The Inverse Laplace Transform......Page 498 Chapter 26: Convolution......Page 510 Chapter 27: Piecewise-Defined Functions and Periodic Functions......Page 524 Chapter 28: Delta Functions......Page 556 Part V: Power Series and Modified Power Series Solutions......Page 574 Chapter 29: Series Solutions: Preliminaries......Page 576 Chapter 30: Power Series Solutions I: Basic Computational Methods......Page 602 Chapter 31: Power Series Solutions II: Generalizations and Theory......Page 646 Chapter 32: Modified Power Series Solutions and the Basic Method of Frobenius......Page 682 Chapter 33: The Big Theorem on the Frobenius Method, with Applications......Page 720 Chapter 34: Validating the Method of Frobenius......Page 744 Part VI: Systems of Differential Equations (A Brief Introduction)......Page 764 Chapter 35: Systems of Differential Equations: A Starting Point......Page 766 Chapter 36: Critical Points, Direction Fields and Trajectories......Page 790 Appendix: Author’s Guide to Using This Text......Page 822 Answers to Selected Exercises......Page 832 Back Cover......Page 864