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. BDH is a hefty book (about 750 pages) and comes with a correspondingly hefty price tag (as of this writing, about $175 on amazon.com); Noonburg’s book is slimmer (about 300 pages of text) and sells for less than one-third the cost of BDH ($50 on amazon as I write this).

Although Noonburg’s book is slim, it covers (and covers well) all of the familiar topics one expects to find in a first semester sophomore-level ODE course, and then some. It also has some interesting features that distinguish it from most of the existing textbook literature, chief among them being a strong emphasis on the dynamical systems approach, which manifests itself in, for example, an early introduction to the idea of a system of differential equations, as well as an early introduction to the concept of the phase line and phase plane for autonomous first and second order ODEs.

Chapter 1 is largely introductory, but takes the usual introductory material farther than is customary in books at this level. It introduces the notion of a differential equation (and a system of differential equations) and the concept of a solution. It is emphasized from the start that sometimes a solution is not obtained as an explicitly defined function but can be found numerically or graphically. Finally, to illustrate the idea of modeling real-world problems with differential equations, several detailed examples involving, among other disciplines, ecology, biology and engineering are given. Real-world modeling examples appear throughout the book, both in the text and in the exercises.

First order equations are the subject of chapter 2. The traditional standard topics (separable equations, exact equations, integrating factors for linear equations, etc.) show up here, but so do other topics that are not usually presented this early (or at all) in a course like this: numerical methods are presented early, as is the graphical approach (looking at slope fields). The upshot is that the student learns early that there is more to the subject of differential equations than mechanical application of rote solution techniques. (And I was pleased to see that this book, like BDH, makes a point of noting that “multiplication by \(dx\)” in separable equations requires some discussion and justification.)

In the next chapter, the equations change from first order to second order. Here again there are the usual discussions of Wronskians, series solutions (done fairly briefly, in about five pages) and the methods of undetermined coefficients and variation of parameters, but there is also an earlier-than-usual introduction to the phase plane of a homogenous equation of the form \(x'' = f(x, x')\), thereby generalizing the concept of phase line that was previously introduced. There is also a brief discussion of numerical methods for second order equations, building on the material from the previous chapter.

The next two chapters are on systems of differential equations. In the first of these two chapters, it is explained how a single *n*^{th} order differential equation can be written as a system of *n* first order ones, and then the remaining part of the chapter addresses systems of linear equations using matrices (such as the matrix exponential). The next chapter, building on the work on phase planes that was begun in chapter 3, is on the geometry of autonomous systems. Linear systems of two equations are considered first (and the phase plane analyzed using the eigenvalues of the 2 × 2 matrix defining the system). This material is then used to analyze, in the next section, nonlinear two-dimensional systems. The chapter ends with a discussion of several important models, discussed in some depth. (The author develops from scratch all the material on matrices and eigenvalues that is necessary for an understanding of these chapters, so a prior course in linear algebra is not a prerequisite for reading this book. Of course, however, prior, or at least concurrent, exposure to this material would be all to the good, as it would be for *any* ODE text.)

The final chapter of the text is a fairly standard look at Laplace Transforms. Though at least one recent book (*Ordinary Differential Equations* by Adkins and Davidson) advocates an early introduction to Laplace Transforms, it remains safe to say that in many introductory courses they are mentioned at the end if, indeed, they are mentioned at all. The placement of this material at the end of the book allows an instructor of such a course to defer or delay this material, as deemed necessary.

Other features also combine to make this an excellent text. The author’s writing style is very clear and should be quite accessible to most students reading the book. There are lots of worked examples and interesting applications, including some fairly unusual ones. There are also numerous exercises, ranging in difficulty from the very routine (verify that such-and-such function is a solution to such-and-such differential equation) to more elaborate student projects, some of which are based on research papers. Some (carefully marked) exercises require computer assistance. Solutions to the odd-numbered problems appear in a 40 page appendix.

This book may not be for everyone, simply because it invokes a different approach than is found in many other books. I do think, however, that the way I first learned differential equations as a student in the early 1970s (which, even then, seemed to me to be no more exciting than an endless set of quadratic formula problems) needs to be changed (and is changing). This book offers a clean, concise, modern, reader-friendly, approach to the subject, at a price that won’t make an instructor feel guilty about assigning it. It is certainly worth a very serious look.

Mark Hunacek ([email protected]) teaches mathematics at Iowa State University.