2nd Edition

# A Course in Ordinary Differential Equations

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**A Course in Ordinary Differential Equations, Second Edition** teaches students how to use analytical and numerical solution methods in typical engineering, physics, and mathematics applications. Lauded for its extensive computer code and student-friendly approach, the first edition of this popular textbook was the first on ordinary differential equations (ODEs) to include instructions on using MATLAB^{®}, *Mathematica*^{®}, and Maple™. This second edition reflects the feedback of students and professors who used the first edition in the classroom.

**New to the Second Edition**

- Moves the computer codes to Computer Labs at the end of each chapter, which gives professors flexibility in using the technology
- Covers linear systems in their entirety before addressing applications to nonlinear systems
- Incorporates the latest versions of MATLAB, Maple, and
*Mathematica* - Includes new sections on complex variables, the exponential response formula for solving nonhomogeneous equations, forced vibrations, and nondimensionalization
- Highlights new applications and modeling in many fields
- Presents exercise sets that progress in difficulty
- Contains color graphs to help students better understand crucial concepts in ODEs
- Provides updated and expanded projects in each chapter

Suitable for a first undergraduate course, the book includes all the basics necessary to prepare students for their future studies in mathematics, engineering, and the sciences. It presents the syntax from MATLAB, Maple, and *Mathematica* to give students a better grasp of the theory and gain more insight into real-world problems. Along with covering traditional topics, the text describes a number of modern topics, such as direction fields, phase lines, the Runge-Kutta method, and epidemiological and ecological models. It also explains concepts from linear algebra so that students acquire a thorough understanding of differential equations.

**Traditional First-Order Differential Equations **

Introduction to First-Order Equations

Separable Differential Equations

Linear Equations

Some Physical Models Arising as Separable Equations

Exact Equations

Special Integrating Factors and Substitution Methods

**Geometrical and Numerical Methods for First-Order Equations **Direction Fields—the Geometry of Differential Equations

Existence and Uniqueness for First-Order Equations

First-Order Autonomous Equations—Geometrical Insight

Modeling in Population Biology

Numerical Approximation: Euler and Runge-Kutta Methods

An Introduction to Autonomous Second-Order Equations

**Elements of Higher-Order Linear Equations **Introduction to Higher-Order Equations

Linear Independence and the Wronskian

Reduction of Order—the Case

*n*= 2

Numerical Considerations for

*n*th-Order Equations

Essential Topics from Complex Variables

Homogeneous Equations with Constant Coefficients

Mechanical and Electrical Vibrations

**Techniques of Nonhomogeneous Higher-Order Linear Equations **Nonhomogeneous Equations

Method of Undetermined Coefficients via Superposition

Method of Undetermined Coefficients via Annihilation

Exponential Response and Complex Replacement

Variation of Parameters

Cauchy-Euler (Equidimensional) Equation

Forced Vibrations

**Fundamentals of Systems of Differential Equations **Useful Terminology

Gaussian Elimination

Vector Spaces and Subspaces

Eigenvalues and Eigenvectors

A General Method, Part I: Solving Systems with Real & Distinct or Complex Eigenvalues

A General Method, Part II: Solving Systems with Repeated Real Eigenvalues

Matrix Exponentials

Solving Linear Nonhomogeneous Systems of Equations

**Geometrical and Numerical Methods for First-Order Equations **An Introduction to the Phase Plane

Nonlinear Equations and Phase Plane Analysis

Bifurcations

Epidemiological Models

Models in Ecology

**Laplace Transforms **Introduction

Fundamentals of the Laplace Transform

The Inverse Laplace Transform

Translated Functions, Delta Function, and Periodic Functions

The

*s*-Domain and Poles

Solving Linear Systems using Laplace Transforms

The Convolution

**Series Methods **Power Series Representations of Functions

The Power Series Method

Ordinary and Singular Points

The Method of Frobenius

Bessel Functions

Appendix A: An Introduction to MATLAB, Maple, and Mathematica

Appendix B: Selected Topics from Linear Algebra

Answers to Odd Problems

References

Index

*A Review, Computer Labs, and Projects appear at the end of each chapter.*

### Biography

**Stephen A. Wirkus** is an associate professor of mathematics at Arizona State University, where he has been a recipient of the Professor of the Year Award and NSF AGEP Mentor of the Year Award. He has published over 30 papers and technical reports. He completed his Ph.D. at Cornell University under the direction of Richard Rand.

**Randall J. Swift** is a professor of mathematics and statistics at California State Polytechnic University, Pomona, where he has been a recipient of the Ralph W. Ames Distinguished Research Award. He has authored more than 80 journal articles, three research monographs, and three textbooks. He completed his Ph.D. at the University of California, Riverside under the direction of M.M. Rao.

Praise for the First Edition:"A Course in Ordinary Differential Equationsdeserves to be on the MAA’s Basic Library List … the book with its layout, is very student friendly—it is easy to read and understand; every chapter and explanations flow smoothly and coherently … the reviewer would recommend this book highly for undergraduate introductory differential equation courses."

—Srabasti Dutta, College of Saint Elizabeth, MAA Online, July 2008"An important feature is that the exposition is richly accompanied by computer algebra code (equally distributed between MATLAB,

Mathematica, and Maple). The major part of the book is devoted to classical theory (both for systems and higher order equations). The necessary material from linear algebra is also covered. More advanced topics include numerical methods, stability of equilibria, bifurcations, Laplace transforms, and the power series method."

—EMS Newsletter, June 2007"This is a delightful textbook for a standard one-semester undergraduate course in ordinary differential equations designed for students who had one year of calculus and continue their studies in engineering and mathematics. The main idea is to focus on the applications and methods of solutions, both analytical and numerical, with special attention paid to applications to real-world problems in engineering, physics, population dynamics, epidemiology, etc. A winning feature of the book is the extensive use of computer algebra codes throughout the text. Assuming that the students have no previous experience with Maple, MATLAB, or

Mathematica, the authors present the relevant syntax and theory for all three programs. This helps students to understand better the theoretical material, use computer support more sensibly, and interpret results of computer simulation properly. Some background material from linear algebra is also provided throughout the text whenever necessary. … The book is nicely written, generously illustrated, and well structured. There are plenty of exercises ranging from drilling to challenging. Additional problems for revision and projects are collected at the end of each chapter. … An excellent blend of analytical and technical tools for studying ordinary differential equations, this text is a welcome addition to existing literature and is warmly recommended as essential reading for a first undergraduate course in differential equations."

—Zentralblatt MATH1931