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Ordinary Differential Equations and Stability Theory: An Introduction

David A. Sánchez
Dover Publications
Publication Date: 
Number of Pages: 
[Reviewed by
William J. Satzer
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This concise introduction to ordinary differential equations was originally published in 1968. The current edition is an unabridged republication by Dover. The book is aimed at advanced undergraduates and beginning graduate students.

The author was unhappy with typical courses of the 1960s that concentrated on special solution techniques for special equations. He wanted to incorporate some stability theory in an introductory course and believed he could do it with students who had “a modicum of knowledge beyond the calculus”. This modicum seems to include some basic linear algebra and a modest level of experience following analytical arguments.

The author begins with statements of existence and uniqueness theorems and a discussion of the maximum interval of existence. Next he introduces linear equations. He begins with a general discussion that treats fundamental solutions, and then goes on to discuss the Wronskian, n-th order homogenous and non-homogenous systems. A chapter on linear equations with constant coefficients follows.

With this background in place the author develops some basic ideas of qualitative analysis using the concepts of the phase plane for autonomous systems of equations. The discussion begins with linear systems and then expands to incorporate nonlinear systems. The author’s treatment here is limited to two-dimensional systems, and he uses phase plane sketches of trajectories to illustrate the basic ideas of stability and asymptotic stability, the varieties of critical points, cycles and limit cycles.

The next chapter ratchets things up considerably. The systems are now non-autonomous and n-dimensional. The author states and proves necessary conditions for linear systems to be asymptotically stable, and then provides two results for asymptotic and uniform stability of nonlinear systems. A few pages are then devoted to a discussion of Liapunov functions and Liapunov stability.

The book concludes with a chapter that provides proofs of the existence and uniqueness theorems stated in the first chapter. The author also states and proves basic results on the continuation of solutions and dependence of solutions on parameters.

In the years since this book was published many new textbooks have appeared that focus on the qualitative analysis of ordinary differential equations and incorporate much of the material included here, and often a good deal more. These newer texts are also more accessible than this one. The combination of the quick pace of the current book and the conciseness of its treatment make it a less desirable choice for an introductory course.

Bill Satzer ( was a senior intellectual property scientist at 3M Company. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.

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