Lectures on Differential Equations

In this introduction to differential equations, the author emphasizes that he wanted to write a book that students would like to read. He is clearly enthusiastic about the subject and has indeed produced a very readable book. The introduction also describes his intention to write in plain language without being too wordy. He feels that an extra word of explanation is often more likely to confuse than illuminate. While theoretical results are mentioned, stated and occasionally proved throughout the book, the focus is on solving differential equations and understanding properties of their solutions.

 

The book is based on three different courses the author has taught. The first part comes from a course for second-year science and engineering students. The second part was used for a graduate course in ordinary differential equations, and the final part in a course on Fourier series, boundary value problems and partial differential equations. The parts all work well enough individually, but it isn’t clear just how instructors might use the book considering the differences in the level of treatment across those three parts.

 

The first part of the book begins with a basic discussion of first-order equations. It starts with linear equations and continues by describing the methods used to find explicit solutions of several kinds of first-order equations. Second-order equations are treated next, and the author treats each variety of second-order equations separately so, for example, there is one section of linear homogeneous equations with constant coefficients where the characteristic equation has two distinct real roots, and another section for the same thing with a repeated real root. This careful step-by-step approach is typical of the first part of the book. The other components of the first part are chapters on infinite series solutions and the Laplace transform.

 

The second part shifts to a more sophisticated treatment of systems of differential equations and moves at a quicker pace. Although the author reviews the essential pieces of matrix and vector manipulations, he is implicitly assuming basic competence with linear algebra. Eigenvalues and the matrix exponential come up right away. Very quickly we get pretty deeply into graduate-level material. The author takes up Floquet theory, states Massera’s theorem on a condition for the existence of periodic solutions, and discusses the Hartman-Grobman theorem for the local linearized behavior of a dynamical system where the hyperbolic equilibrium is at the origin. Then he moves on quickly to controllability and observability for linear systems. Nonlinear systems and the topics of Lyapunov stability, limit cycles and the Poincaré-Bendixson theorem are treated in the final chapter of the second part.

 

Partial differential equations and Fourier series make up the third part. The level here is intermediate between those of the first and second parts. The treatment is mostly standard although there are a couple of particularly good applications. 

 

Unlike many recent texts, this one spends little time on numerical computation. Numerical solutions are discussed briefly in the first chapter, and a short chapter at the end describes the capabilities of various software systems, but there is no attempt to incorporate these in the text or exercises.

 

The book is full of interesting pieces. It excels with problems and applications and demonstrates considerable enthusiasm for the subject. Just how the book might work for a course is not so clear. The significant disparity in level between the various parts makes its use for a text difficult. Readers seeking to learn differential equations by selectively working their way through a succession of topics might find it appealing.

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Bill Satzer (bsatzer@gmail.com), now retired from 3M Company, spent most of his career as a mathematician working in industry on a variety of applications ranging from speech recognition and network modeling to optical films. He did his PhD work in dynamical systems and celestial mechanics.