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Ordinary Differential Equations and Linear Algebra: A Systems Approach

Todd Kapitula
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
, on

Many books that treat ordinary differential equations with linear algebra don’t do well by linear algebra. It’s not necessarily that the exposition is poor; it’s just that linear algebra is treated… well, like an accessory. This book takes it seriously, and in so doing, does well by the student. Students who take a course based on books like this are usually second or third year undergraduates in engineering and related fields, and the course is often required.

The author notes that the substantive linear algebra component makes it easier for the students when discussing solutions of systems of linear ODEs to focus more on the ODE aspects and less on the corresponding algebraic operations. He argues that this also enables a more extensive treatment of linear systems of ODEs and more interesting examples. (An additional benefit is that those engineering students that go on to a course often called something like “Systems and Signals” will be especially well-prepared.)

The book begins with a long chapter on linear algebra — almost a hundred pages — before differential equations are even mentioned. The topics here are mostly standard. The author does not talk about linear transformations, but he does define vector space, subspaces, and null space and state theorems about equivalent conditions that guarantee existence and uniqueness of solutions for linear systems. Many theorems and lemmas are proved or a sketch of the proof is given. The chapter concludes with four “case studies” — applications of the ideas of the chapter. These include a digital signal filter with transfer function (anticipating Laplace transform work later in the book), and three discrete dynamical system models that have convenient matrix formulations.

With the next two chapters the author introduces scalar first-order linear differential equations and then systems of such equations. In each case the treatment begins with motivating problems, briefly touches on existence and uniqueness theorems for the initial value problem, and then takes up the structure of solutions. The book has a clear sense of logical flow so that the treatment of systems of ODEs falls into place naturally following the path set down in the linear algebra chapter. Once again there are some very well designed case studies. These involve, for example, tank-mixing problems, lead levels in the human body, and models of infectious disease.

Following this the author discusses linear higher order scalar ODEs along with variation of parameters and undetermined coefficients solution methods. He also uses this section to pull together examples of mass-spring systems (undamped, damped, overdamped and coupled). The author seems to understand very well that engineers and physicists build and retain their knowledge of differential equations around examples just like these.

It is a bit unusual to treat discontinuous forcing and the Laplace transform in a book at this level as the author does in the next chapter. Nonetheless it is once again appropriate and provides an excellent background for engineering and physics students (and not bad for the mathematics major either). The treatment here naturally leads to a discussion of transfer functions and convolutions. A final chapter adds some odds-and-ends like separation of variables methods and power series solutions.

The book is a bit heavy on engineering and physics examples for a course that might include students with primary interests in biology or economics. Yet the treatment is so smooth that it would work for a broader collection of students as long as the instructor balances it with examples and applications from other areas. The twenty case studies are more than just good examples; they encourage students to go beyond simply finding solutions to understanding the dynamical behavior that they represent.

Plenty of exercises at a variety of levels are provided. The author also proposes group projects in most chapters. One of these in an early chapter invites students to explore Google’s PageRank algorithm.

This is an attractive text that is well worth a look, especially for courses with a predominance of engineering students. I wish that it had been available when I last taught a course like this.

Bill Satzer ( 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.