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Ordinary Differential Equations: A Brief Eclectic Tour

Mathematical Association of America
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Reading this book is like having a series of delightful faculty room conversations with a genial colleague who happens to be a very experienced researcher and teacher in the field of differential equations. Even when you disagree with the ideas expressed, you can learn something by listening carefully. As Sánchez himself puts it, "the book is more conceptual than definitive, and more lighthearted than pedagogic."

As precursors of this volume, certainly there are the three books and many research papers Sánchez has written or co-authored; but, in addition, we should consider his insightful, thought-provoking review of six textbooks in ODEs that was published in the Monthly (April, 1998, pp. 377-383). In Sánchez's review, which influenced my own book and which I recommend enthusiastically, his theme is that ODE textbooks have become corpulent, swollen by unnecessary topics and digressions that detract from the beauty and elegance of the subject. After analyzing the treatments of several key topics, he concludes his essay by the admonition "Maybe the watchword should be 'Whoa!' and not 'Wow!'." The book under review is the result of further thinking on Sánchez's part and focuses on what he sees as the blurring or loss of the analytic and geometric foundations of the subject.

The book is arranged into five chapters, following a traditional ordering of topics: Solutions, First Order Equations, Insight Not Numbers [numerical methods], Second Order Equations, and Linear and Nonlinear Systems. Each chapter contains revealing comments and illuminating examples, insights that even an instructor who has taught ODEs many times before may not have. There is an annotated list of references and an index. This is not a textbook, but a valuable supplement to any teacher's repertoire. There are delights awaiting even the most jaded of instructors. Bright students who have finished an ODE course will deepen their knowledge by working their way through this book.

In the very first chapter, the author discusses existence and uniqueness (both in terms of a partial derivative criterion and a Lipschitz condition) and provides an intuitive treatment of the continuation of the interval of existence to a maximum interval of existence. In treating the basic separable equation, to which many of us give short shrift, Sánchez provides a pedagogically sound recommendation to expand the treatment of the equation dx/dt = f(t)g(x) into three separate cases: (A) g(x) = 1 identically, (B) f(t) = 1 identically, and (C) neither A nor B. He introduces the qualitative analysis of ODEs in the second chapter, applying it to the logistic equation with harvesting, first discussed by Sánchez and Brauer in 1975.

Sánchez acknowledges a weakness for the Riccati ODE, "the author's favorite equation," despite an earlier warning to avoid "exotic special ODEs." He proceeds to give an informative overview of this equation and its applications.

The book is replete with gems of analysis and common sense. The blurb on the back cover of the book singles out several novel items, such as the existence of periodic solutions of first-order equations and the use of Gronwall's Lemma to prove a key result in the analysis of almost linear systems, but almost every page is filled with interesting and important insights. His third chapter derives its title and philosophy from the famous quote of Richard Hamming: "The purpose of computation is insight not numbers." The author is critical of books with titles such as "Differential Equations with ", concerned that such texts emphasize number crunching and pretty pictures over conceptual understanding. Sánchez recommends using the Euler and Improved Euler methods to introduce the theory of numerical approximation and the analysis of error; but he suggests jumping to the RKF45 scheme for practical computation. Above all, he encourages the development of insight.

Surprisingly, he comments on the "formidable task" involved in calculating or estimating the period in an autonomous two-dimensional system with a periodic solution. However, I have found (specifically, in treating the Lotka-Volterra equation and Zeeman's model of the human heartbeat) that using a CAS implementation of an RKF45 algorithm solves this problem easily.

In making his points, Sánchez writes with grace and a sense of humor. He makes several references to "the little gnome inside the computer." In omitting a discussion of RLC circuits in his treatment of second-order equations, the author admits that he "doesn't know an ohm from an oleaster." His analysis of mechanical systems with friction asks the reader to imagine that "the cylinders of your Harley Davidson's shock absorbers were filled with Perrier, Mazola, or Elmer's Glue." With tongue in cheek, Sánchez sees on the silver screen a "mathematical monster movie — Call the National Guard — the Annihilator has escaped!."

No book escapes a reviewer's grasp without mention of a few typos and some cavils. These are minor for this book. For example, on p. 9, Kamke's classic book is given as Differential Gleichgungen... instead of Differentialgleichungen... (as on p. 128) and umlauts are missing from the rest of the title on both pages. On p. 30, tenth line from the bottom, "left" should be "right" and on the twelfth line from the bottom, the last subscript should be 3. In stressing the importance of the formula for the complex exponential (p. 65), the author fails to mention Euler's name. While omitting electrical circuits as examples, Sánchez shows a predilection for control theory problems that may be beyond the average instructor's (or student's) ken.

Throughout the book, the author is critical of the excessive emphasis on numerical analysis, linear algebra, and technology in many differential equations courses and suggests alternative approaches. The reader seeking a balance should have both this book and a copy of the recent MAA volume Revolutions in Differential Equations handy. The last part of Sánchez's FINALE (p.125) deserves to be reproduced in its entirety:

   The subject is
         and NOT
             Linear Algebra
             Numerical Analysis
             Computer Science

Henry Ricardo ( is Professor of Mathematics at Medgar Evers College of The City University of New York and Vice-Chair for Four-Year Colleges of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002.

Date Received: 
Thursday, June 6, 2002
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David A. Sánchez
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Wednesday, January 16, 2008