Differential Equations with Applications and Historical Notes

My first course in differential equations was a failure. It may have been my fault, of course. I did learn how to solve linear differential equations, and I remember the endless proof of existence and uniqueness of solutions, particularly the theorem that explained how the local solutions could be assembled into a solution that was valid in as large a region as possible. But very little stuck.

A few years later I found myself needing to teach the basics of differential equations to a class of engineering students, part of their fourth semester calculus course. It was at that point that I ran into George Simmons’s Differential Equations with Applications and Historical Notes and fell in love with it.

Simmons’s book was very traditional, but was full of great ideas, stories, and illuminating examples. Consider, for example, the first chapter, “The Nature of Differential Equations.” After the usual general remarks, Simmons provides examples: families of curves, growth and decay, chemical reactions, falling bodies, and (amazingly) the brachistochrone. (The third edition adds some material from probability theory.) It’s hard going for a beginner, but it immediately establishes the main point: this stuff plays a central role in the physical sciences. Differential equations are powerful.

Simmons’s “historical notes” often appear in footnotes. Some are fairly tame stuff, summaries of the lives of the mathematicians whose names are attached to various equations, methods, or theorems. But every once in a while it gets more exciting. For example, after saying that a certain “surprising property is really quite natural” (on page 275 of this edition) the footnote says “Those readers who are blessed with indomitable skepticism, and rightly refuse to accept assurances of this kind without personal investigation, are invited to consult…” Fantastic! Similarly, after an account (on p. 172) of why the sum of the reciprocals of the squares is \(\pi^2/6\), there is the footnote: “The world is still waiting — more than 200 years later — for someone to discover the sum of the reciprocals of the cubes.” This is a charming book whose author has a clear personality and doesn’t hide his preferences.

Some of my readers may be wondering what the value of \(\zeta(2)\) is doing in a book on differential equations. That is another characteristic of this book: Simmons ranges far and wide through mathematics, bringing in whatever seems useful and relevant to the matter at hand.

The book is not without its faults, as I discovered when I tried to use it with Colby students (more than twenty years ago!). Simmons is mostly interested in physics and engineering, and both his problems and his examples reflect that fact. My class was full of budding economics students who didn’t find this material all that interesting.

The book is definitely old-school. There is some material on qualitative theory of solutions, but there are whole chapters on power series solutions, special functions, Fourier series, and the Laplace transform. Even the calculus of variations gets a chapter. Existence and uniqueness get their chance in the penultimate chapter, followed by a chapter on numerical methods that seems unchanged since the second edition. This is not a competitor for Blanchard-Devaney-Hall or Noonberg.

Some years ago, an attempt was made to update Simmon’s book. The result was published as Differential Equations: Theory, Technique, and Practice, by Simmons and Steven Krantz. Alas, much of the charm of the original disappeared in the new version. So it is good news that CRC has brought back the original book in a third edition. I compared it to the second edition and decided that the changes are mostly minor additions dealing with topics Simmons enjoys. Most importantly, the author’s unique personality shines through.

I wouldn’t use Simmons as the main textbook for a differential equations course. Given the right student, it might make a great source for an independent study. It is definitely worthwhile to have this classic back.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College.