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Few books on Ordinary Differential Equations (ODEs) have the elegant geometric insight of this one, which puts emphasis on the qualitative and geometric properties of ODEs and their solutions, rather than on routine presentation of algorithms.

There are dozens of books on ODEs, but none with the elegant geometric insight of Arnol'd's book. Arnol'd puts a clear emphasis on the qualitative and geometric properties of ODEs and their solutions, rather than on theroutine presentation of algorithms for solving special classes of equations.Of course, the reader learns how to solve equations, but with much more understanding of the systems, the solutions and the techniques. Vector fields and one-parameter groups of transformations come right from the startand Arnol'd uses this "language" throughout the book. This fundamental difference from the standard presentation allows him to explain some of the real mathematics of ODEs in a very understandable way and without hidingthe substance. The text is also rich with examples and connections with mechanics. Where possible, Arnol'd proceeds by physical reasoning, using it as a convenient shorthand for much longer formal mathematical reasoning. This technique helps the student get a feel for the subject. Following Arnol'd's guiding geometric and qualitative principles, there are 272 figures in the book, but not a single complicated formula. Also, the text is peppered with historicalremarks, which put the material in context, showing how the ideas have developped since Newton and Leibniz. This book is an excellent text for a course whose goal is a mathematical treatment of differential equations and the related physical systems.

Although there is no lack of other books on this subject, even with the same title, the appearance of this new one is fully justified on at least two grounds: its approach makes full use of modern mathematical concepts and terminology of considerable sophistication and abstraction, going well beyond the traditional presentation of the subject; and, at the same time, the resulting enhancement of mathematical abstractness is counterbalanced by a constant appeal to geometrical and physical considerations, presented in the main text and in numerous problems and exercises. In the terms of mathematical approach, the text is dominated by two central ideas: the theorem on rectifiability of a vector field (which is equivalent to the usual theorems on existence, uniqueness, and differentiability of solutions) and the theory of one-parameter groups of linear transformations (equivalent to the theory of linear autonomous systems). The book also develops whole congeries of fundamental concepts--like phase space and phase flows, smooth manifolds and tangent bundles, vector fields and one-parameter groups of diffeomorphisms--that remain in the shadows in the traditional coordinate-based approach. All of these concepts are presented in some detail, but without assuming any background on the part of the reader beyond the scope of the standard elementary courses on analysis and linear algebra The theory of ordinary differential equations is one of the basic tools of mathematical science.