You are here

Differential and Integral Equations

Peter J. Collins
Oxford University Press
Publication Date: 
Number of Pages: 
[Reviewed by
Henry Ricardo
, on

In 362 pages, the author conveys a good deal of information about first- and second-order ordinary and partial differential equations and their connections to integral equations. The material was developed in classes at the College of William and Mary and at Oxford. The style is somewhat formal, but the topics are handled well, with examples, clear diagrams, and exercises to illuminate and supplement the text. There are no references to using technology and no answers/solutions to exercises appear at the back of the book.

This text assumes a previous course in ordinary differential equations, although some basic material is included in the Appendix. (These days, Chapter 15, on Phase-Plane Analysis, may be considered review material as well.) A very brief sketch of important results in single and multivariable calculus is given in Chapter 0. Linear algebra is introduced where required and an introduction to complex function methods is provided at key places in the text. The author provides different paths through the material via a schematic presentation of the book’s contents and helpful discussions of the interdependence of the chapters. The book ends with a 48-item bibliography.

In contrast to most modern first courses in differential equations, virtually all applications of the material are concerned with classical problems of mathematical physics. For instance, only as an exercise in the last chapter does Volterra’s predator-prey model appear.

As the publisher’s blurb asserts, this is “analysis for applications” and this well-written text is suitable for physicists, economists, chemists, and any student interested in acquiring a working knowledge of applied analysis.

Henry Ricardo ( is Professor of Mathematics at Medgar Evers College of The City University of New York and Secretary of the Metropolitan NY Section of the MAA. His book, A Modern Introduction to Differential Equations, was published by Houghton Mifflin in January, 2002; and he is currently writing a linear algebra text.
How to use this book
1. Integral equations and Picard's method
2. Existence and uniqueness
3. The homogeneous linear equation and Wronskians
4. The non-homogeneous linear equation: Variations of parameters and Green's functions
5. First-order partial differential equations
6. Second-order partial differential equations
7. The diffusion and wave equations and the equation of Laplace
8. The Fredholm alternative
9. Hilbert-Schmidt theory
10. Iterative methods and Neumann series
11. The calculus of variations
12. The Sturm-Liouville equation
13. Series solutions
14. Transform methods
15. Phase-plane analysis
Appendix: The solution of some elementary ordinary differential equations