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 the routine 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 start and 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 hiding the 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 historical remarks, which put the material in context, showing how the ideas have developed 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.

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