This is a textbook for a one-semester introductory course on differential equation. For a book in this category, it is on the shorter side. It consists of only seven chapters, the first of which is terminology, and the second of which is first order equations.
It is after this that the author has to make real decisions. First, he discusses numerical approximations (the Euler method and the Runge-Kutta method) before moving any higher. Second and higher order equations are treated similarly as in other textbooks, though the use of the word "eigenvalue" for the roots of the characteristic equation struck this reviewer as unusual.
The most significant difference between this book and competing textbooks is in chapter five, on systems of linear differential equations. In several competing textbooks, that material is covered only in one section. The reason for that is that at many universities, students are required to take this class before they take linear algebra, and not the other way around. In this book, the author includes the necessary definitions of eigenvalues, eigenvectors, and linear dependency, though most of the chapter (all but one section) is devoted to the case of 2x2 matrices.
Then comes a chapter on the Laplace transform, which is fairly usual, and the book ends with a chapter on nonlinear differential equations, and the Poincaré-Lyapunov theorem. The latter is not found in many competing textbooks.
There are about 30 exercises per section, and the odd-numbered ones have their numerical answers included at the end. There are three appendices, one on calculus topics such as improper integrals and power series, one on complex numbers, and one on linear algebra. These will all be welcomed by students. On the other hand, the fact that power series are only mentioned here, and not in a chapter discussing power series solutions of differential equations, will be a negative for many instructors.
Finally a word on pricing. As I am typing this, the book costs $129.50 on amazon.com. There are competing textbooks that have enough material for two semesters that are $20 to $30 cheaper.
Mikós Bóna is Associate Professor of Matheamtics at the University of Florida.