Ordinary and Partial Differential Equations for the Beginner

Despite the title, this is not an introduction to differential equations for students fresh out of calculus. Instead, the author implicitly assumes that the reader has a much stronger background. Already five pages into the book we encounter a section titled “Auxiliary results from functional analysis” and this includes the Banach Fixed Point and Arzelà-Ascoli theorems. So it’s not a book for the kind of beginner you might imagine. 

 

Who, then, are the intended readers? The author notes in his introduction that he used the material for a course in a Masters in Science program. His expectations are implicit but clear: students coming to this book already have a strong background in analysis and competence with basic linear algebra. Yet the primary aim of the book seems to be to teach students to find solutions of a variety of ordinary and partial differential equations. That does not mean that theory is neglected. In some respects, there is more careful attention to the formal analytic aspects of differential equations than one sees in some graduate texts. Theorems with proofs appear throughout - but finding solutions to differential equations seems to be the priority. A majority of the many exercises in the book ask the student to find the solution of a particular equation. 

 

Most of the topics treated in the book are standard, but the manner in which they are examined is rather unusual. Despite his emphasis on finding solutions, his exposition attends very carefully to analytical details. The author begins with existence and uniqueness of solutions and proceeds through this very rigorously. He then introduces solutions of special equations of the usual basic types (e.g., separable, exact, implicit, and first-order linear) early in the book, and then immediately treats more advanced equations like the Bernouilli, Ricatti, Lagrange and Clairut equations. 

 

Linear differential equations in more than one real or complex dimension are taken up next, followed by higher-order differential equations. The author’s treatment of the latter includes brief discussions of the Laplace and Fourier transforms, as well as power series solution methods. 

 

The presentation of partial differential equations emphasizes two primary themes. One is first-order partial differential equations and Cauchy problems. The author presents the theory of characteristics and shows how it can be used to solve linear and nonlinear first-order equations. The second theme focuses on classical methods used to solve higher-order equations and related Cauchy problems. Here there is less emphasis on theory and more on the available solution methods.

 

The text is different in many respects from comparable books at this level. It has no graphics at all – no plots, graphs or figures. Numerical methods and numerical solutions of differential equations are also not addressed. Nor does the qualitative analysis of solutions make an appearance. The book depends strictly on formal analytic techniques all the way through. 

 

This book could be suitable for an introductory graduate course. Despite what seems to be an unusual focus and an occasionally idiosyncratic treatment, it is clear and rigorous. Nevertheless, it is far enough from what has become a standard approach to the subject that it is more likely seen as useful supplementary reading.

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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 and material science. He did his PhD work in dynamical systems and celestial mechanics.