Ordinary Differential Equations

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 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.

In terms of concrete applications, the book introduces the pendulum equation at the very beginning, and the efficacy of various concepts and methods introduced throughout is subsequently tested by applying them to this example. Thus, the section on first integrals leads to the law of conservation of energy; the theorem on differentiation with respect to a parameter finds application in the "method of small parameters"; and the theory of linear equations with periodic coefficients merges naturally with the study of parametric resonance. This geometrical and physical specificity is made still more vivid through the inclusion of 259 line drawings and 260 exercises in which other examples are taken up.

Reader's review

This has to be one of the most amazing math books I've ever read. Arnol'd seems to do the impossible here - he blends abstract theory with an intuitive exposition while avoiding any tendency to become verbose. By the end of Arnol'd, it's hard not to have a deep understanding of the way that ODEs and their solutions behave.
Arnol'd accomplishes this feat through an intense parsimony of words and topics. Everything in this book builds on the central theme of the relations between vector fields and one-parameter groups of diffeomorphisms, and the topics are illustrated (and often motivated) almost exclusively through problems in classical mechanics, most notably the plane pendulum. Almost no solution techniques are given in this book - expect no mention of integrating factors or Bessel functions. One of the main reasons that the book does so much without bogging down is that the mathematical formalism is minimal and terse - proofs are often one or two lines long, merely mentioning the conceptual justification of a result without detailed, formal constructions.
But the result of this parsimony is that Arnol'd is a very difficult book. To understand every detail and to be able to attempt every problem, I think, basically requires a math degree - lots of linear algebra (for his monumental 116-page chapter on linear systems), a solid background in analysis and topology, and a bit of differential geometry and abstract algebra are prerequsites for a full understanding. (I found the section on the "topological classification of singular points," in particular, nearly incomprehensible with my thin chemistry-major math background.) There are foibles, too, including proofs that satisfy the requirements for some theorem or definition without actually stating what theorem or definition is now applicable. One can detect some mild arrogance in places (after an arduous two-page proof, he mentions "As always in proving obvious theorems, it is easier to carry out the proof of the extension theorem than to read through it.") Also, a few typos can be found here and there, which sometimes result in confusion.
One very curious thing about Arnol'd is that my most brilliant math-major friends find it impenetrable, whereas I know biologists who got through it with no problem. So I guess that, for a mere mortal, reading Arnol'd demands a willingness to have a feel for a big picture without worrying about every epsilon and delta.
So grab a copy of this book, let it flow, and learn about ODEs. It's well worth the effort.