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A Textbook on Ordinary Differential Equations

Shair Ahmad and Antonio Ambrosetti
Publication Date: 
Number of Pages: 
Unitext 73
[Reviewed by
William J. Satzer
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This text on ordinary differential equations is pitched somewhere between an introductory course for undergraduates in the US and a first course at the graduate level. The level of sophistication is a bit too high for most undergraduates, yet the breadth is probably insufficient for a typical graduate course. So it falls pretty much in the middle. Nevertheless it has its attractions, either for an advanced undergraduate course or a supplemental text.

One of the authors indicates that his motivation was to provide a more affordable textbook. This is an admirable goal and perhaps a good enough reason for another differential equations book, yet I was a little skeptical: after all, this book is not cheap either. Then I was shocked to find some standard texts selling for more than $200. Especially with that in mind, this book is a good value. It is well organized, clearly written and includes topics that could nicely supplement an undergraduate course.

After an introductory chapter on first order linear equations, the authors introduce the general theory of first order equations in the second chapter. Here they state existence and uniqueness theorems for the Cauchy initial value problem (with proofs in an appendix to the chapter). The treatment is succinct, reasonably well motivated, and rigorous. Existence and uniqueness for higher order equations and systems are treated a few chapters later.

The next several chapters address first order nonlinear equations, second and higher order equations and systems of first order equations. The nonlinear equations include those few for which closed-form solutions exist: equations separable, exact, solvable via integrating factor, or of Bernoulli type. An appendix to the chapter looks at singular solutions and Clairaut equations.

Beyond this is an assortment of topics that could nicely supplement a more typical introductory undergraduate course. These include boundary value problems and Sturm-Liouville eigenvalue theory, the Laplace transform, series solutions and Bessel functions, and a little bit of stability theory.

The authors do not discuss computer methods at all. They are firmly of the belief that students at this level should concentrate on analytical techniques. There is however a short appendix that describes the numerical methods of Euler and Runge-Kutta.

This is an appealing mid-level text. It has many exercises with a significant number of them focused on the solution or analysis of particular equations or systems. The authors evidently put high priority of the development of skills like this.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.