Ordinary Differential Equations: An Introduction to the Fundamentals (Textbooks in Mathematics)

Description

The Second Edition of Ordinary Differential Equations: An Introduction to the Fundamentals builds on the successful First Edition. It is unique in its approach to motivation, precision, explanation and method. Its layered approach offers the instructor opportunity for greater flexibility in coverage and depth. Students will appreciate the author’s approach and engaging style. Reasoning behind concepts and computations motivates readers. New topics are introduced in an easily accessible manner before being further developed later. The author emphasizes a basic understanding of the principles as well as modeling, computation procedures and the use of technology. The students will further appreciate the guides for carrying out the lengthier computational procedures with illustrative examples integrated into the discussion. Features of the Second Edition: Emphasizes motivation, a basic understanding of the mathematics, modeling and use of technology A layered approach that allows for a flexible presentation based on instructor's preferences and students’ abilities An instructor’s guide suggesting how the text can be applied to different courses New chapters on more advanced numerical methods and systems (including the Runge-Kutta method and the numerical solution of second- and higher-order equations) Many additional exercises, including two "chapters" of review exercises for first- and higher-order differential equations An extensive on-line solution manual About the author: Kenneth B. Howell earned bachelor’s degrees in both mathematics and physics from Rose-Hulman Institute of Technology, and master’s and doctoral degrees in mathematics from Indiana University. For more than thirty years, he was a professor in the Department of Mathematical Sciences of the University of Alabama in Huntsville. Dr. Howell published numerous research articles in applied and theoretical mathematics in prestigious journals, served as a consulting research scientist for various companies and federal agencies in the space and defense industries, and received awards from the College and University for outstanding teaching. He is also the author of Principles of Fourier Analysis, Second Edition (Chapman & Hall/CRC, 2016). Contents Preface PART I The Basics 1 The Starting Point: Basic Concepts and Terminology 2 Integration and Differential Equations PART II First-Order Equations 3 Some Basics about First-Order Equations 4 Separable First-Order Equations 5 Linear First-Order Equations 6 Simplifying Through Substitution 7 The Exact Form and General Integrating Factors 8 Review Exercises for Part of Part II 9 Slope Fields: Graphing Solutions Without the Solutions 10 Numerical Methods I: The Euler Method 11 The Art and Science of Modeling with First-Order Equations 12 Numerical Methods II: Beyond the Euler Method PART III Secondand Higher-Order Equations 13 Higher-Order Equations: Extending First-Order Concepts 14 Higher-Order Linear Equations and the Reduction of Order Method 15 General Solutions to Homogeneous Linear Differential Equations 16 Verifying the Big Theorems and an Introduction to Differential Operators 17 Second-Order Homogeneous Linear Equations with Constant Coefficients 18 Springs: Part I 19 Arbitrary Homogeneous Linear Equations with Constant Coefficients 20 Euler Equations 21 Nonhomogeneous Equations in General 22 Method of Undetermined Coefficients (aka: Method of Educated Guess) 23 Springs: Part II (Forced Vibrations) 24 Variation of Parameters (A Better Reduction of Order Method) 25 Review Exercises for Part III PART IV The Laplace Transform 26 The Laplace Transform (Intro) 27 Differentiation and the Laplace Transform 28 The Inverse Laplace Transform 29 Convolution 30 Piecewise-Defined Functions and Periodic Functions 31 Delta Functions PART V Power Series and Modified Power Series Solutions 32 Series Solutions: Preliminaries (A Brief Review of Infinite Series, Power Series and a Little Complex Variables) 33 Power Series Solutions I: Basic Computational Methods 34 Power Series Solutions II: Generalizations and Theory 35 Modified Power Series Solutions and the Basic Method of Frobenius 36 The Big Theorem on the Frobenius Method, with Applications 37 Validating the Method of Frobenius PART VI Systems of Differential Equations (A Brief Introduction) 38 Systems of Differential Equations: A Starting Point 39 Critical Points, Direction Fields and Trajectories 40 Numerical Methods III: Systems and Higher-Order Equations Appendix Author’s Guide to Using This Text Answers to Selected Exercises Index