Ordinary Differential Equations

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.