Ordinary Differential Equations: Introduction and Qualitative Theory

Cronin’s book, with a prerequisite of the first semester of advanced calculus and a semester of linear algebra, is aimed at advanced undergraduates and beginning graduate students — undergraduates would have to be rather mature mathematically to appreciate this book. It is a classic treatment of many of the topics an instructor would want in such a course, with particular emphasis on those aspects of the qualitative theory that are important for applications to mathematical biology. In general, the number of exercises at the end of the chapters is adequate, although the last two chapters have no exercise sets. There is a lengthy appendix on topics from real analysis and metric spaces, especially the concept of topological degree.

The book starts with a thorough treatment of the classical existence and uniqueness theorems, illustrating a variety of proof methods. The first chapter concludes with examples and exercises showing the application of these results to predator-prey equations (Volterra), the Hodgkin-Huxley equations, the Field-Noyes model for the Belousov-Zhabotinsky reaction, and the Goodwin equations for a chemical reaction system.

As the TOC indicates, most of the rest of the book deals with systems of differential equations (especially two-dimensional systems, with emphasis on the Bendixson theory) and stability results. The treatment is mathematically rigorous throughout. Some potential users of this book may be discouraged by the “brief and uneven” discussion of boundary value problems and the Sturm-Liouville theory. Another criticism is that, although Lyapunov functions are defined and various stability/instability criteria are proved, there is no explanation of how to construct such functions.

A nice feature of this edition is an extended and unified treatment of the perturbation problem for periodic solutions. The author demonstrates that use of the averaging method to study periodic solutions is a special case of a method of Poincaré.

The book concludes with seven and a half pages of references, mostly to the classic literature. I think it’s significant that there are only two books on this list with publication dates later than 2000 — and both deal with applications. The latest edition of the popular text by Boyce and DiPrima referenced is the 4th, although we are now up to the 9th edition.

Overall, this book is a solid graduate-level introduction to ordinary differential equations, especially for applications. It is reminiscent of the classic texts of Birkhoff and Rota and of Coddington and Levinson, rather than, say, the recently updated book by Hirsch, Smale, and Devaney.

Henry Ricardo ([email protected]) has retired from Medgar Evers College (CUNY), but continues to serve as Governor of the Metropolitan NY Section of the MAA. He is the author of A Modern Introduction to Differential Equations (Second Edition). His linear algebra text was published in October 2009 by CRC Press.