A Course in Differential Equations with Boundary-Value Problems

This voluminous introduction to differential equations has material sufficient to support a variety of undergraduate courses from basic to more advanced levels. It appears to have been designed to offer instructors considerable flexibility in selecting topics to meet the goals of a particular course. It also provides options to accommodate academic schedules that range from individual quarters to multiple semesters.

The authors follow a traditional approach to ordinary differential equations with a few wrinkles and a touch of a more modern approach. They begin with first order linear equations and continue to treat linear equations and linear systems completely before introducing nonlinear systems. Numerical methods are introduced early in the book. Parallel computer lab exercises are provided at the end of each chapter in Matlab, Mathematica, and Maple. An appendix provides short tutorials for each of these software packages. The authors also introduce some qualitative analysis early, largely in the form of direction fields.

All the linear algebra that’s needed is presented early in the chapter that introduces systems of differential equations. The authors explore qualitative analysis of systems by introducing the phase plane and illustrating its use with the harmonic oscillator system. They go on to describe the various possible equilibria that one can encounter and expand that to consider nonlinear systems and bifurcations in systems that depend on a parameter. A series of examples with epidemiological and ecological models demonstrate the usefulness of these ideas in applications.

Laplace transforms and power series methods for solving differential equations are treated in mostly standard ways. A chapter on boundary value problems (BVPs) touches on several areas including two-point BVPs, Sturm-Liouville theory, orthogonal functions, and Fourier series. It is essentially a quick survey that gives students a taste of the subject. Likewise, the chapter on partial differential equations is a relatively short introduction to the subject concentrating on the heat, wave and Laplace equations.

Besides the exercises and computer labs that are presented in each chapter, the authors also provide projects that explore some subjects in greater detail or introduce new related material. These include applications (like signal processing, integral equations, and stochastic processes) as well as more advanced mathematical material (e.g., hypergeometric functions, Chebyshev polynomials).

The book’s flexible organization offers instructors a lot of options for courses of various lengths, levels, and choices of computer software. This offers some obvious advantages. But it also makes for a very long book and one that may not be well matched to any particular course. Yet it is thoughtfully assembled, well written and attentive to the needs of students.

This text has much in common with another book with some of the same authors. The first edition of that book was reviewed here. The current book, itself a second edition, shares eight identical chapters and two appendices with the second edition of the prior book. The book reviewed here adds the two chapters on boundary value problems and partial differential equations.

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.