This book presents a modern treatment of material traditionally covered in the sophomore-level course in ordinary differential equations. While this course is usually required for engineering students the material is attractive to students in any field of applied science, including those in the biological sciences.

The standard analytic methods for solving first and second-order differential equations are covered in the first three chapters. Numerical and graphical methods are considered, side-by-side with the analytic methods, and are then used throughout the text. An early emphasis on the graphical treatment of autonomous first-order equations leads easily into a discussion of bifurcation of solutions with respect to parameters.

The fourth chapter begins the study of linear systems of first-order equations and includes a section containing all of the material on matrix algebra needed in the remainder of the text. Building on the linear analysis, the fifth chapter brings the student to a level where two-dimensional nonlinear systems can be analyzed graphically via the phase plane. The study of bifurcations is extended to systems of equations, using several compelling examples, many of which are drawn from population biology. In this chapter the student is gently introduced to some of the more important results in the theory of dynamical systems. A student project, involving a problem recently appearing in the mathematical literature on dynamical systems, is included at the end of Chapter 5.

A full treatment of the Laplace transform is given in Chapter 6, with several of the examples taken from the biological sciences. An appendix contains completely worked-out solutions to all of the odd-numbered exercises.

The book is aimed at students with a good calculus background that want to learn more about how calculus is used to solve real problems in today's world. It can be used as a text for the introductory differential equations course, and is readable enough to be used even if the class is being "flipped." The book is also accessible as a self-study text for anyone who has completed two terms of calculus, including highly motivated high school students. Graduate students preparing to take courses in dynamical systems theory will also find this text useful.

*Solutions manuals available upon request. Please contact: Carol Baxter at cbaxter@maa.org.*

### Table of Contents

Preface

Sample Course Outline

1. Introduction to Differential Equations

2. First-order Differential Equations

3. Second-order Differential Equations

4. Linear Systems of First-order Differential Equations

5. Geometry of Autonomous Systems

6. Laplace Transforms

A. Answers to Odd-numbered Exercises

B. Derivative and Integral Formulas

C. Cofactor Method for Determinants

D. Cramer’s Rule for Solving Systems of Linear Equations

E. The Wronskian

F. Table of Laplace Transforms

Index

About the Author

### About the Author

**V.W. Noonburg**, better known by her middle name Anne, has enjoyed a somewhat varied professional career. It began with a B.A. in mathematics from Cornell University, followed by a four-year stint as a computer programmer at the knolls Atomic Power Lab near Schenectady, New York. After returning to Cornell and earning a Ph.D. in mathematics, she taught first at Vanderbilt University in Nashville, Tennessee and then at the University of Hartford in West Hartford, Connecticut (from which she has recently retired as professor emerita). During the late 1980s she twice taught as a visiting professor at Cornell, and also earned a Cornell M.S. Eng. degree in computer science.

It was during the first sabbatical at Cornell that she was fortunate to meet John Hubbard and Beverly West as they were working on a mold-breaking book on differential equations (*Differential Equations: A Dynamical Systems Approach, Part I*, Springer Verlag, 1990). She also had the good fortune to be able to sit in on a course given by John Guckenheimer and Philip Holmes, in which they were using their newly written book on dynamical systems (*Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields*, Springer-Verlag, 1983). All of this, together with being one of the initial members of the C-ODE-E group founded by Bob Borrelli and Courtney Coleman at Harvey Mudd College, led to a lasting interest in the learning and teaching of ordinary differential equations. This book is the result.

### MAA Review

All of us have our favorite books in various areas of mathematics, and when it comes to elementary differential equations my favorite was *Differential Equations* by Blanchard, Devaney, and Hall (hereinafter BDH). There were several things that I particularly liked about the book, which struck me as somewhat less “cookbooky” than the typical sophomore ODE text at this level. I particularly appreciated, for example, the emphasis in BDH on the dynamical system approach, which struck me as a good way to learn the subject, and I also liked the fact that BDH addressed certain little things that other books often gloss over: for example, in the discussion of variables-separable equations, BDH acknowledges that “multiplying” the equation \( dy/dx = f(x)g(y) \) by \(dx\) is something that raises some concerns, and discusses a justification for the process.

I think, however, that the book under review has now edged out BDH as my favorite basic ODE text. As will be shortly noted, the things that I like about BDH are also present here, but this book also remedies what I thought was the one significant problem with using BDH as a text: its price. Continued...