Elementary Differential Equations

This is an introduction to ordinary differential equations for undergraduates. The intended audience is broad and includes students majoring in mathematics, applied mathematics, computer science, and engineering as well as those in the physical or social sciences. From the beginning, the author is clear that he wants to follow a different path. He argues that little has changed in the way differential equations have been taught over the last fifty years. His contention is that computing should be a much more significant component of an introductory course.

The author’s motivations for his approach are based on two arguments. The first is that students who go on to technical careers that require the use of differential equations will be expected to use computational methods to find numerical solutions. He also argues that students now have little exposure to nonlinear differential equations and even less to nonlinear systems because explicit solutions are rarely possible. Computer software makes numerical solutions of such equations and systems relatively straightforward, so he says students should be taught how to use it.

Despite this emphasis on computation much of the material in the book looks familiar. The author says he wants to maintain a balance between theory, computation, and applications and generally he does so. He handles the initial value problem for first-order equations in the usual way and offers several applications. However, the idea of numerical solutions is introduced much earlier than in comparable texts. The book proceeds to introduce the usual topics in roughly the usual order: n-th order linear differential equations, systems of first-order equations, and so on with a full set of applications at each stage. The difference, after the first two chapters, is that numerical solutions are routinely demonstrated in examples and called for in exercises.

The author notes that any of the widely used computer software packages (Maple, MATLAB, Mathematica, and the like) could be used and the text is independent of whatever software might be chosen. But he does provide his own software on the book’s website that he uses to generate solutions and graphs for his examples as well as answers to selected exercises. The student choosing to use other software would need guidance in selecting the appropriate routines for a given exercise since none is offered in the text.

A chapter on the Laplace transform makes a surprise appearance here, given the author’s emphasis on computation. The presentation is nicely done and it includes a discussion of associated computational tools. The author shows how the Laplace transform can be used to solve nonhomogeneous linear differential equations with constant coefficients where the inhomogeneity is a discontinuous function, a time-delay function, or an impulse function. This feature is of particular value in engineering applications.

It is perhaps a bit disconcerting to think of turning students loose with computer tools to solve differential equations without some serious discussion of how those tools can run amuck. The author does offer some “what can go wrong” advice. He concentrates on making sure that students attend to the requirements of the existence, uniqueness and continuation theorems and “conduct a thorough mathematical analysis” for each initial value problem. Mostly he discusses making sure that the intervals for which solutions are sought are appropriate for the form of the equations and avoid singularities. Of course, there are quite a few subtler pitfalls but they are not discussed here. It may be too much to expect students to do the kind of general mathematical analysis the author suggests. Without more tools than students have at their disposal, that can be very difficult.

Quite a number of books with titles including words like “Ordinary Differential Equations with X” are available now where X is MATLAB. Mathematica, Maple, or something similar (this one, for example), but they are not as focused on computational solutions as the current book is.

This is a carefully assembled text with plenty of exercises that could work as an introductory text, but it would probably require more support than usual from the instructor.

Bill Satzer (bsatzer@gmail.com) was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.