500 Examples and Problems of Applied Differential Equations

Yes, there are almost 500 examples and problems from ordinary differential equations (ODE's) here, all dealing with applications. (Actually 368 problems and 109 examples.) Two of the authors (Agarwal and O’Regan) have previously published textbooks on ODE's. They decided that these books needed many more examples and exercises with applications to make them complete. This turned out to be a bigger project than they expected and led to the current book.

 

It’s not just about problems and exercises. All of the usual topics typically included in undergraduate ODE courses are represented. The authors provide background material for each area that describes the minimum required theory (with theorems but without proofs). They also demonstrate some techniques and then offer a number of examples and exercises. The topics begin with first-order linear equations. progress through the usual topics, and end with nonlinear boundary value problems.

 

The authors claim no particular originality for their examples and exercises. They say that their goal is simply to identify the applications and present them as simply and clearly as possible. Their secondary goal is to inspire readers to pursue research inspired by the applications they present.  The range and variety of examples and applications are remarkable. These come from physics, chemistry, geophysics, biology, medicine, economics, and several different engineering fields. While a number of them – especially those in the first few chapters - will be familiar to those who have taught ODE courses, quite a few are new. 

 

Some of the most valuable features of the book are the examples and exercises in areas like the Runge-Kutta method, power series methods, stability theory, and boundary value problems. Texts that treat these subjects often provide few exercises with applications.  The authors have provided solutions for many of the exercises of the “find the solution” type, and hints for at least some of the “show that” type. The level of difficulty generally increases throughout the book. Things get tougher in the chapters on boundary value problems, especially the nonlinear ones.  Most of the book is accessible to undergraduates, though the material on nonlinear boundary value problems would be particularly challenging. The authors have aimed the book at mathematics, physics and engineering students, but the first several chapters have some very good biological and medical applications.   Each chapter comes with a very complete list of references that provide more background information and ideas for further work. Many of the examples have pointers to the literature for those interested in pursuing them further.

 

This book is probably most suitable as a supplement to a regular text or as a source that instructors could use for additional exercises. It would also be possible for a student with a special interest in applications to use the text for self-study, as long as a more standard book was available to fill in proofs and reinforce the important theoretical details.

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Bill Satzer (bsatzer@gmail.com), 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 Ph.D. work in dynamical systems and celestial mechanics.