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Ordinary differential equations and linear algebra : a systems approach
Kapitula T., SIAM, Philadelphia, PA, 2015. 312 pp. Type: Book (978-1-611974-08-9)
Date Reviewed: Jun 6 2016

When I was an undergraduate, the differential equations class had a poor reputation. It was said to be difficult, with lots to learn, grinding homework, and really awful exams. It was certainly so for me, but then over the years, with more math classes and a stint at teaching physics, I started to appreciate how important the topic really is, and how interesting it can be. This book is a very nice presentation and probably would have helped me as a student to appreciate the topic more quickly.

As well as covering differential equations, it also presents enough linear algebra so that students without linear algebra experience should be able to handle the course. This is not a complete text in linear algebra and doesn’t even really touch on the numeric aspects of linear algebra, but since that is not needed for the rest of the text, the coverage is good and appropriate.

The book consists of six chapters: (1) “Essentials of Linear Algebra,” covers enough linear algebra to prepare the reader for what is to come (thankfully, with minimal numeric matrix manipulation); (2) “Scalar First-Order Linear Differential Equations”; (3) “Systems of First-Order Linear Differential Equations”; (4) “Scalar Higher-Order Linear Differential Equations”; (5) “Discontinuous Forcing and the Laplace Transform”; and (6) “Odds and Ends,” which includes separation of variables and power series solutions.

Most of the text is theoretical, but with some basis in practice. Motivating examples are given at the beginning of each chapter, and there is a nice set of exercises for each section and group projects for each chapter. These group projects are longer, usually motivated in some practical problem and requiring a reasonable amount of thought as well as understanding of the material.

Readers are encouraged to use computer algebra systems--mentioned in particular are Sage and Wolfram Alpha, though in a few places MATLAB may be used for numerical solutions. It is good to see free systems used, and equally good that the text does not take any time to teach the reader how to use these. While neither Sage nor Wolfram Alpha are without their problems, they’re both good systems and it is good that students are exposed to them. There is some supporting material for Sage, MATLAB, and Mathematica on the book’s website.

Since computer algebra systems are used, students do not need to spend time grinding away on numerical problems (a few numerical problems are helpful in getting started, but too many may leave students frustrated and bored); instead, they can spend time exploring solutions and methods, even posing problems for themselves. This also makes it much easier to produce reasonable graphics, which perhaps as much as anything can aid in understanding.

The book is 300 pages long; 100 pages make up the linear algebra chapter. It reads a bit dense and is likely suitable for students who are comfortable with mathematics in general. For those less so, the theoretical bias may be a bit off-putting; however, there is little in here that is not well explained and accessible, so the book could certainly be used as a text in a class where numeric methods are emphasized.

The book has many examples, most of which are based in real-world problems (mixing problems, glucose regulation, disease models), and most of the group exercises are grounded in real-world examples.

There is an index, but it is very short and not very complete. Indexing the problems and examples a bit more thoroughly (they are indexed, but not very helpfully) could be a big help in a future edition.

Reviewer:  Jeffrey Putnam Review #: CR144472 (1608-0550)
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