Has technology and software made a difference in mathematics pedagogy or textbooks? Maybe yes, maybe no, but I think the difference is less than predicted 15 years ago. The second edition of *Differential Equations: An Introduction with Mathematica*, by Clay Ross, is a case in point. This textbook is surprisingly similar to the gray textbook I learned from, when universal personal computing was not even a gleam in a Silicon Valley entrepreneur's eye.

This text instructs students in solving and using differential equations with both paper-and-pencil techniques and the *Mathematica* symbolic manipulation program. *Mathematica* commands assist the solution of the differential equations, streamlining tedious computations while paralleling the paper-and-pencil steps. Ross acquaints the reader with *Mathematica* commands by example with a minimum of syntax explanation. *Mathematica* use is interactive, avoiding programming and user-created multi-step commands. Occasionally the text introduces "black-box" *Mathematica* commands which solve differential equations directly, but this typically just mentions such commands exist and checks or compares the solutions to the lengthier step-by-step computation. *Mathematica* also functions in this book as a way for students to check answers and in fact the book does not have solutions to problems in the back. Instead, Ross explicitly directs students to check their answers by modifying the *Mathematica* examples. The book comes with a CD with notebooks providing further examples of using *Mathematica*.

The differential equations topics covered are the usual sequence of first-order equations, standard applications of first-order equations, higher-order linear constant coefficient equations, applications to oscillations and electrical circuits, Laplace transform methods, series solutions and linear systems with phase-plane diagrams. A quick survey of linear algebra appears in Chapter 2, and then the language of linear algebra describes the solution space of differential equations throughout the book. Second-order and higher-order equations are in the same chapter in subsequent sections. The techniques of variation of parameters and undetermined coefficients are treated equally for non-homogeneous equations, and undetermined coefficients is approached through the method of annihilators.

In spite of the availability of *Mathematica*, numerical and graphical methods are not emphasized. Direction fields and Euler's method appear in the first chapter but do not reappear in the chapter on applications of first-order equations. The fourth-order Runge-Kutta algorithm is mentioned as a higher-order extension of Euler's method along with *Mathematica*'s **NDSolve** command. In the chapter on applications of first-order equations, the only numerical applications are a numerical integration for a specific solution and **NDSolve** is used for a Ricatti equation. In two very short sections in the chapters on applications of second-order equations and on systems, Ross derives a nonlinear equation for planetary motion and mentions the Lotka-Volterra equations, but neither is explicitly solved numerically. Plotting solutions with *Mathematica* is not prominent, for example, in the chapter on first-order equations the text only demonstrates plotting a family of solutions of a Bernoulli equation, and plotting the solutions of the Clairaut equation. Plotting of solutions is more prominently displayed in the section on damped second-order equations. No *Mathematica* plotting commands are in the chapter on systems, although the many figures have obviously been prepared with *Mathematica*. I did not receive the accompanying CD for review, maybe the examples I am looking for are there.

The index does not include *Mathematica* commands. There are several places in the book where bad typography mangled some mathematical symbols (for example rendering the ≠ sign into "\ ="), and this could puzzle some students.

So what do we have here? This is a good, thorough standard textbook with good explanations. However, this textbook is very similar to the textbook I learned elementary differential equations from, with the addition of *Mathematica* commands to automate calculation and further illustrate the solution techniques. I have seen the future, and it works just like the past.

Steven R. Dunbar (

sdunbar@unl.edu) teaches at the University of Nebraska - Lincoln and is MAA Director of Mathematics Competitions.