Friday, April 18, 2014

MAA Books Beat: Extraordinary Book on Ordinary Differential Equations

Written by Steve Kennedy, MAA Books Beat is a column written for MAA FOCUSExtraordinary Book on Ordinary Differential Equations appears in the April/May 2014 issue.

Mathematics changes slowly; the mathematics curriculum changes even more slowly. A few years ago, to celebrate the tercentenary of L'Hôpital's Calculus, some colleagues and I read it seminar-style. The most striking thing about the experience was that his table of contents looked shockingly similar to our departmental calculus syllabus. That being said, in the 30 years I've been teaching collegiate mathematics, there is one course in the undergraduate math curriculum that has changed dramatically–the course in ordinary differential equations (ODE).

These changes are rooted in the calculus reform movement of the late 1980s and early 1990s, the easy access to powerful computing and visualization, and the rise of dynamical systems theory and its accompanying qualitative point of view.

The calc reform movement taught us to explain everything from graphical, numerical, and symbolic perspectives. Computing, of course, made it possible to do this in effective ways, especially the numerical and graphical bits. Dynamical systems provided entirely new ways of thinking about the evolution and bifurcation of systems.

The New on View


All these changes are fully on view in Virginia (“Anne”) Noonburg's Ordinary Differential Equations from Calculus to Dynamical Systems, newly released by the MAA. Noonburg has a distinguished record of research in dynamical systems, especially concentrating on equations that model biological systems. You clearly see these intellectual interests in this book.


In the first chapter we learn (a simplified version) of an ordinary differential equation model of nerve cell activity in the brain alongside the logistic growth model for populations (and models for current in a circuit and terminal velocity of skydivers).

It is nice to see the early emphasis on modeling as the purpose of differential equations. This clear emphasis makes natural the focus on qualitative and geometric features of solutions, which is a theme throughout the book.

Chapter 5 is particularly focused on qualitative methods. It begins with one of my favorite classifications, the organization of two-dimensional constant coefficient linear systems in the trace-determinant plane, which is just a beautiful scheme.

After a discussion of equilibrium type and linearization, the rest of the chapter is taken up by unusually deep qualitative analysis of several models. These models include the Van der Pol equation, a competing species model, the Wilson-Cowan equations that model neuronal behavior, and a predator-prey model. As I said, all of these models are analyzed in greater-than-usual depth. There are deep and interesting bifurcation analyses for all of these.

Meaty Exercises


I should also mention that this chapter is especially well stocked with meaty exercises. If you are looking to give your students an experience that captures something of what actually happens when a scientist models a phenomenon–and, one could argue, this should be one of the purposes of an ODE course–then you, and your students, will be rewarded by careful study of this chapter.

This is a pretty terrific book for a first ODE course. It has a nice modern flavor and covers exactly the topics that I like to cover, and from the point of view that I think is the way to do it. The book contains many, many great exercises. Your students will find the text very readable–so readable that you should be able to use this book in a flipped classroom course.


Steve Kennedy, senior acquisitions editor for MAA Books, can be contacted at kennedy@maa.org.

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