X-factor resides in big picture

The textbook team of Andrew King and John Billingham has reunited, now joined by Steven Otto, to produce another attractively packaged, potential bestseller. Aimed at undergraduate students taking general courses on differential equations, Differential Equations: Linear, Nonlinear, Ordinary, Partial is divided into two sections.

The first 200 pages cover linear differential equations, ordinary and partial. The approach is straightforward and clear, including Frobenius methods, Fourier series analysis and solutions by Laplace transform. Green functions, Sturm-Liouville systems, Legendre and Bessel functions are also discussed at length. The standard classification, properties and complex variable methods for second-order linear partial differential equations (PDEs) are present, although condensed into a disappointingly short 25 pages. This brevity is perhaps the weakest point of the text, but doubters should not dismiss the book before taking into account the considerable distinguishing features offered by the second part.

Covering a variety of topics including existence and uniqueness of solutions, phase-plane methods for higher-order nonlinear equations, group theoretic methods, instability, bifurcation theory, optimal control and chaotic systems, each topic is self-contained and well explained. Seven appendices provide more basic material for the weaker (or forgetful) students. A fifth of the entire book is devoted to a detailed account of asymptotic methods. This is clear and comprehensive enough to be used by itself for an independent course on asymptotics.

Derived from road-tested lecture courses, the book is suitable for use in most university mathematics degrees, especially those with an applied element. But the explicit nature of the first section also allows it to be used on higher-level undergraduate engineering or physics courses. Examples from these areas are included, varying from elementary technical exercises to those embedded in research contexts.

Readers may immediately seek to compare the merits of this text with, for example, the admirable Applied Partial Differential Equations by John Ockendon et al . The quick answer is that the two books serve different purposes. The book under review covers large portions of essential undergraduate material other than PDEs. Therefore, it cannot cover all aspects of PDEs in as great a theoretical detail as Ockendon et al .

Depending on the level of students, courses devoted exclusively to PDEs might prefer Ockendon. Those teaching more general differential equations or methods courses should certainly consider King et al .

Christopher Howls is senior lecturer in applied mathematics, University of Southampton.