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Differential Equations

Paul Blanchard, Robert L. Devaney, and Glen R. Hall
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The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

I taught Differential Equations in 2013–14, both Fall and Spring semesters. Not really my thing, of course, but the people who normally teach that course were both on sabbatical and someone had to step in. It had been some twenty years since I last taught the course, so I had to do a lot of catching up.

When I taught the course in the mid-90s, I used the famous older textbook by Simmons, Differential Equations with Applications and Historical Notes. It’s a great book, charming and intelligent, but it clearly shows its MIT roots in that most of the examples and applications come from physics and engineering. At the end of the course, several of my students reminded me that I was teaching at a liberal arts college, so that a wider range of examples might be desirable.

I knew as well that a great deal had changed in the field. Like many older books, Simmons emphasizes equations that can be solved in closed form. I wanted to center my course on a more qualitative approach, recognizing that these days graphical and numerical methods tend to dominate. Of course, I also talked to my colleagues, and one of them showed a decided preference for the book under review. So a year later, having taken the plunge and survived, here I am to tell you what the water is like.

Blanchard, Devaney, and Hall teach at Boston University, so that the book reflects their long experience teaching the introductory differential equations course there. They emphasize the dynamical and qualitative points of view from the first page on: in this book, the variable is always called t and almost always stands for time. The reader is reminded repeatedly, with what appear to be real-life examples, that “solving” does not just mean “finding a formula.” The book comes with software that is easy to use and that can be used to make concepts come to life. (It is unclear to me whether this software is free or not; the book comes with an access code but the web site does not seem to actually ask for that code.) The exercises are generally quite good. Cengage provides an online solutions manual that can be used to produce printed solutions for students.

Let me point out some differences between this book and those I used when I was a student:

  • The term “autonomous” appears early and is very prominent.
  • By contrast, the word “linear” appears later and only becomes really central in chapter three.
  • Exact first-order equations aren’t even mentioned. Integrating factors are only mentioned in the context of linear first-order equations, which do not come first.
  • Phase lines and phase planes appear early, and bifurcations do as well.
  • The logistic equation is one of the major examples, appearing throughout the book in various guises.
  • There is much emphasis on setting up equations and understanding what they say.
  • Understanding solutions is more important to the authors than writing them down.
  • There are no problems having to do with determining curves from the properties of their tangents (this is perhaps the loss I most regret).

There are a lot of good concrete examples. I particularly enjoyed one about controlling the vibration of a washing machine, which is quite complex (a good thing for students to see) and really puts the theory to work.

Two downsides stuck out for me. The first has to do with mathematical pre-requisites: the authors assume no linear algebra. This means, for example, that during their treatment of (two-dimensional) linear systems of differential equations the authors need to introduce eigenvalues and eigenvectors from scratch. We make the linear algebra course a pre-requisite, so I felt a little impatient with these sections of the book. On the other hand, some of my students seem to have appreciated the chance to review. In the end, the main effect of this limitation is that higher-dimensional problems are mostly ignored. Even three-dimensional systems get little attention.

The other, more significant, downside has to do with style: this is a very chatty book. The first time I used it, my scheduling turned out all wrong because I tended to assume that if the book spent ten pages on something I should dedicate some serious class time to it. The second time around I was much more efficient, letting students read for themselves and streamlining things a lot. This got me much further into the book and resulted in a better course. Anyone using this book as a text needs to constantly ask “Is this something worth spending class time on?”

Overall, this is a pleasant book to work with. I am convinced it has the right take on the subject: given that Mathematica can find closed-form solutions when they exist, there is little point to teaching an old-style “find the formula” kind of course. Plus, the qualitative theory is much more fun. You get the chance to tell stories about ecological catastrophes explainable by bifurcation theory, to relate the topology of the plane to properties of solutions, and to introduce students to deterministic chaos. If the price doesn’t scare you away (and you like the dynamical systems point of view) it should certainly be on the short list for textbook selection.

Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME. He doesn’t know if he will ever teach differential equations again.

Modeling via Differential Equations. Analytic Technique: Separation of Variables. Qualitative Technique: Slope Fields. Numerical Technique: Euler's Method. Existence and Uniqueness of Solutions. Equilibria and the Phase Line. Bifurcations. Linear Equations. Integrating Factors for Linear Equations.
Modeling via Systems. The Geometry of Systems. Analytic Methods for Special Systems. Euler's Method for Systems. The Lorenz Equations.
Properties of Linear Systems and the Linearity Principle. Straight-Line Solutions. Phase Planes for Linear Systems with Real Eigenvalues. Complex Eigenvalues. Special Cases: Repeated and Zero Eigenvalues. Second-Order Linear Equations. The Trace-Determinant Plane. Linear Systems in Three Dimensions.
Forced Harmonic Oscillators. Sinusoidal Forcing. Undamped Forcing and Resonance. Amplitude and Phase of the Steady State. The Tacoma Narrows Bridge.
Equilibrium Point Analysis. Qualitative Analysis. Hamiltonian Systems. Dissipative Systems. Nonlinear Systems in Three Dimensions. Periodic Forcing of Nonlinear Systems and Chaos.
Laplace Transforms. Discontinuous Functions. Second-Order Equations. Delta Functions and Impulse Forcing. Convolutions. The Qualitative Theory of Laplace Transforms.
Numerical Error in Euler's Method. Improving Euler's Method. The Runge-Kutta Method. The Effects of Finite Arithmetic.
The Discrete Logistic Equation. Fixed Points and Periodic Points. Bifurcations. Chaos. Chaos in the Lorenz System.
A. Changing Variables.
B. The Ultimate Guess.
C. Complex Numbers and Euler's Formula.