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Exploring ODEs

Lloyd N. Trefethen, Ásgeir Birkisson, and Tobin A. Driscoll
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Allen Stenger
, on

This is a very interesting text in ordinary differential equations (ODEs), that can be used as a supplement for an introductory course or as a follow-on course. It starts with the radical proposal that we are not going to solve any ODEs explicitly. Instead we will use the computer to find and plot numerical solutions, and we will tweak the ODE and its initial or boundary value conditions to see qualitatively how that affects the solution.

The book is based on MATLAB and the Chebfun package, that can do this kind of solving and plotting with no trouble. I think MATLAB is not essential; I ran a number of the examples in Mathematica (using the NDSolve function) and it usually worked well. The non-oscillatory boundary-value problem in Chapter 5 failed because Mathematica did not get a good numeric approximation (it did warn you that it was not a good solution). Chebfun has an especially concise syntax, which makes it easy to use. But in any case you usually are just entering one line of differential equation (with initial or boundary-value conditions) and one line of plot command, so any computer system that can solve ODEs numerically should be usable.

Each chapter is fairly narrow and focused, and usually deals with a particular kind of solution behavior rather than a particular form of equation. For example, Chapter 8 deals with resonance, and we experiment with the forcing function to see how the amplitude builds up when the forcing function’s frequency is close to the resonant frequency. The book is chock-full of practical examples of ODEs. Each chapter considers one real problem in detail, and then has another dozen or so in the exercises. The book takes the ODEs as given and does not do any modeling to obtain them.

Very Good Feature: Each chapter has a section “Our Favorite Reference”. It gives a relevant reference work for the type of differential equation covered in that chapter (usually a book, sometimes a journal paper) along with a few sentences explaining why it is their favorite. About half of these are ODE books, but the other half are from a wide variety of application areas.

The hardcover book is reasonably priced, as textbooks go, and you can also download the book in PDF form for free from Trefethen’s web site.

Bottom line: a unique and valuable book, that will give you a much better understanding of how differential equations work than most courses would.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.