Written by Steve Kennedy, MAA Books Beat is a column written for MAA FOCUS. Extraordinary Book on Ordinary Differential Equations appears in the April/May 2014 issue.
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
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.
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 email@example.com.