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A Modern Introduction to Differential Equations

Henry Ricardo
Academic Press
Publication Date: 
Number of Pages: 
[Reviewed by
Bill Satzer
, on
This is a text for a one-semester course in ordinary differential equations. It is the third edition of a book first published in 2009.  See our review of the previous edition.
The structure of the book has not changed significantly since the first edition. As in previous editions, the concepts are developed in the context of dynamical systems using a combination of qualitative, quantitative, and geometric methods. The author has done a bit of reorganization and added new examples, exercises, and figures. The biggest changes are additions of new material, most of it on the more advanced side.
The new material includes an expanded treatment of bifurcations, for first-order equations and then more broadly for linear and nonlinear systems. The author has also added an optional section on limit cycles and the Hopf bifurcation. There is a particularly good application example here with the Van der Pol oscillator. This edition also includes a longer section on the Hartman-Grobman theorem (called the Lyapunov-Poincaré theorem in the previous edition) with some good examples. This is an unusual topic for a basic one-semester course but now seems to be treated more commonly at an undergraduate level.
As with many books at this level, a background in linear algebra is not assumed. When I have taught a course like this (usually with many engineering students who never take a linear algebra course), I found incorporating linear algebra in this context awkward and difficult to integrate smoothly. Here the author handles that relatively gracefully.
One notable feature of the book is that it provides essentially no discussion of supporting software. The author assumes that students have access to a computer algebra system and possibly some specialized software for graphing and numerical approximation. Otherwise, he offers no software instruction and says that students should follow their instructor’s direction. This works, more or less, because the text shows detailed results of calculations and presents plenty of graphs of phase portraits, solution curves, and the like. This does have the merit of focusing attention on differential equations and not the software.
This is an attractive book, designed for readability and well suited for an introductory course. It has a broad collection of worked-out examples and exercises that span application areas in biology, chemistry and economics as well as physics and engineering.
Bill Satzer (, now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films and material science. He did his PhD work in dynamical systems and celestial mechanics.