I would use *Introduction to Differential Equations Using Sage *in an undergraduate introductory Differential Equations (DE) course if my students did not have strong mathematical skills and if no other software, such as *Mathematica*, was available.

There are of course many DE textbooks on the market. In analyzing the virtue of any textbook it is useful to have a global viewpoint on DE which could serve as a yardstick to measure textbooks. From my point of view, any intro DE course must cover the following six topics, each of which has distinct techniques: 1) First order equations 2) second (and higher) order equations, 3) Systems of equations, 4) Series solutions, 5) Numerical methods and 6) Modeling. If time and student skills permit, the following four topics can be added: 7) Laplace transforms, 8) geometric analysis (isoclines, phase planes and stability), 9) Partial Differential equations (e.g. Fourier transforms and the heat equation), and 10) use of computer systems.

A text like *Elementary Differential Equations and Boundary Value Problems, *by William Boyce and Richard DiPrima, is considered excellent and a standard because it covers in detail all ten of the above topics and has numerous problems. Consequently, if I was teaching a student population with strong skills and had access to a good computer system such as *Mathematica*, I would use Boyce and DiPrima. In fact, Joyner and Hampton acknowledge previously using Boyce and DiPrima in their courses though they indicate that “for the most part we taught things a bit differently.”

With this background we can now assess the Joyner and Hampton book. It has five virtues which an instructor should carefully consider when choosing a text.

1) **Assumed student skills**: It is very frustrating, when teaching weaker students, to have to review, say, linear algebra, Taylor series, or convergence of power series. Typically, one must refer to other textbooks and photocopy exercises from them. How refreshing therefore, to see this review in the book proper. Joyner and Hampton carefully spend several sections reviewing basic linear algebra, power series and Taylor series. The instructor need not refer to any former textbooks to review these topics. Problems are also provided.

2)** Software friendly**: The book uses the __Sage__ computer software system. Computer code is carefully gone over in the text. Moreover, *Sage* is free. This is helpful in situations where universities do not have licenses for software systems. It is also helpful for students who work and are frequently off campus with a consequent lack of access to software. By using *Sage*, students can always spend time and complete computer-based assignments.

3) **Novel Applications**: Joyner and Hampton present the classical applications such as Newtonian Mechanics, mixing problems and cooling problems. They also present new and novel applications, such as modeling battles using Lancaster’s equation, Romeo and Juliet, and modeling spreads of disease (Zombie attack).

4) **Exercises**: The number of exercises is adequate. Furthermore, as indicated above, there are exercises in the review sections of prerequisite material such as linear algebra, power series and Taylor series. The exercises include software exercises (specific to *Sage*).

5) **More advanced topics**: Joyner and Hampton do have entire sections on material sometimes not covered in an introductory course. More specifically, there are sections on the Laplace transform as well as on partial differential equations, Fourier series and the heat equation.

I have three criticisms of the book (which hopefully will be remedied in future versions).

**Chapter Organization**: I mentioned the Boyce and DiPrima approach above. DE covers ten specific topics, each involving distinct methods. Logically, there should therefore be ten distinct chapters. Joyner and Hampton have four chapters: First order, second order, systems and partial DE. There is a certain logic for example of covering numerical methods at the same time you teach the exact methods for first order equations or for covering power series and Laplace transforms as part of the coverage of second and higher order equations. The pace is certainly leisurely, with a proper number of sections. However, I feel that four chapters are too few; pedagogically, distinct topics should each have their own chapter, as this enables the student to focus on each topic distinctly.

**Geometric methods**: I was a bit surprised that Joyner and Hampton devote only one section — four pages and three exercises! — to the increasingly important topic of isoclines and direction fields. I do hope that future editions of the book will expand on this coverage, especially since their book uses computer systems.

**Bibliography**: Too many items in the bibliography refer to __Wikipedia__. Some of the items are biographical and perhaps they should be in the chapters proper rather than in the bibliography. If the authors wish to give an introduction to a topic, it is preferable to cite Wolfram than Wikipedia. In fairness to the authors, some of the references are nice. For example, I was pleased to find two free e-textbooks on linear algebra which were well written and have plenty of exercises.

Russell Jay Hendel, [email protected], holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.