This is a cookbook of how to solve ordinary and partial differential equations. It’s not a theorems or proofs book, but is devoted to showing by example that particular types of equations have particular types of functions as their solutions. Most of the book consists of worked examples, but it also has clear narratives explaining why we use the methods we do. The book also includes a fair amount about numerical and graphical methods to handle the cases where we cannot find an explicit solution, although this material is dated today. This is a Dover unaltered reprint of the 1961 English Universities Press edition.

The book has an especially strong discussion of the special functions of mathematical physics, such as the Bessel functions and the various types of orthogonal polynomials. It also provides a lot of detail about hypergeometric series. It has hundreds of exercises; many of these are bare equations, but many are presented as problems in mechanics, electricity or physical chemistry where the student must develop and solve the differential equation (usually the relevant physical laws are quoted). The back of the book gives the solutions for all problems (that is, the function that satisfies the differential equation) but does not show how it was obtained. (A few problems are not given solutions; these are the ones where the function is stated in the exercise and the task is to show it is a solution.)

This is primarily an ODE book. The PDE portion is skimpy and mostly deals with particular equations that are useful in physics, rather than general methods. It covers separation of variables and a little bit about Fourier series and Fourier transforms.

The bulk of the work has held up over time. The most dated part is the numerical and graphical part, which was written before computers were widely available and assumes you will do all the work by hand or with the help of a desk calculator. The only numerical methods covered are the relaxation method and the perturbation method, along with brief descriptions of several extrapolation methods. The graphical portion is mostly about phase-plane diagrams.

Most engineering students today would learn differential equations from a comprehensive engineering handbook, such as Kreyszig’s *Advanced Engineering Mathematics*, so those, rather than other differential equations books, are probably the competition for the present book. Kreyszig actually has more pages (and a lot more information) about differential equations than the present book has in total, and the information is more up-to-date. On other hand, Kreyszig is ridiculously expensive at just under $300, while the present book is a bargain at $20.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is allenstenger.com. His mathematical interests are number theory and classical analysis.