Carrier and Pearson is a very interesting book. It is quite concise, and it severely restricts its scope in order to achieve depth - it covers little besides exact, approximate, and (some) numerical solution techniques for first-order and second-order linear ODEs. Its main property is that it takes a simple, highly heuristic approach to these ODEs and is full of rather tough problems, which take up about as much space in the text as the exposition. (Indeed, the authors claim that 78% of the value of the book lies in doing the problems.) It is moderately difficult; I would not recommend C&P to most students as a first text.



The most attractive feature of C&P is indeed its problems, which make vigorous use of the solution techniques presented, introduce the reader to new techniques at times, and give some insight into exactly when those techniques are applicable. If one were to do all of the problems (something I certainly have not done), he or she would really know this subject! C&P's exposition is high in quality, too; when working on a problem set in my second course on ODEs, I would often turn to C&P and find a clean, short, understandable explanation of the tool I needed.



Its drawbacks largely stem from the same philosophy that makes it such a nice book about low-order linear ODEs. Its treatment makes heavy use of basic algebraic manipulation, and it avoids theory almost entirely. C&P eschews the vector-space ideas that clarify topics like the solution of nonhomogeneous linear equations. The simplifying emphasis on basic algebra also obscures the generalization of things like the Wronskian to higher-order systems, and it certainly prevents an even remotely rigorous treatment of Sturm-Liouville systems or eigenfunction expansions. The lack of a modern, geometric view of ODEs (cf. Arnol'd, Ordinary Differential Equations) does not help the student in later making a transition to qualitative considerations of nonlinear ODEs, and it prevents an appreciation of how special the standard linear solution techniques are. C&P also avoids complex analysis; while this is good for a student who has not studied complex variables, the lack of complex analysis means that C&P only inverts Laplace transforms with tables (not contour integration) and has no treatment at all of Fourier transforms. Also, the emphasis on problems means that some very important techniques (like variation of parameters) show up only in the problems.



Whether or not this book is a worthwhile buy depends on the reader. It is great for learning a number of applied-mathematical techniques by the Socratic method. However, it fails as an encyclopedic reference, a mathematician's textbook, or a gateway to nonlinear dynamical systems theory. At any rate, it is a unique book, and at least a portion of science/engineering students would benefit from it.