This is a very good book on Differential Equations. It is the kind of book I would use in the classroom as well as recommend to a student for independent study. I can see it used as textbook for a course in Differential Equations, year-long if Calculus and Linear Algebra background needs to be reinforced, or in a one semester course for math majors.
The book does not include series solutions, the Laplace transform, or numerical methods, so it may not be immediately useful to engineering majors. On the other hand, it builds the theory of Differential Equations, and it does it well. Floquet theory is in Chapter 2, autonomous systems are discussed in Chapter 3, and Chapter 4 contains perturbation methods. It may seem that these topics are out of place in a textbook for a first class in Differential Equations, but through their choice of examples and exercises the authors make the topics flow at a natural pace.
There are nice small side topics, such as the construction of sine and cosine functions and the proof of the basic trigonometric identities based solely on the existence and uniqueness theorem, as well as less usual topics, such as factorization theorems.
Chapters 5 to 8 take the reader through Sturm-Liouville problems, the calculus of variations, higher order systems, nonlinear ODE’s, and classical existence and uniqueness theorems. These topics are sufficiently rich for each one of them to be the subject of a separate book. The present text deals with them to the point where substantial theorems can be discussed, and at the same time leaves the reader wanting to find out what more can be said. The references are appropriate and point to specific pages in the classic texts (Coddington and Levinson, Hartman, etc.).
I believe that instructors would enjoy teaching from this book, and that students would be able to study from it (either through a class or independently) at a good pace. And they would learn a lot about differential equations.
Florin Catrina is Assistant Professor of Mathematics at St. John's University in Queens, New York.