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.
Vladimir Igorevich Arnold (alternative spelling Arnol'd, Russian: Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician. While he is best known for the Kolmogorov–Arnold–Moser theorem regarding the stability of integrable systems, he made important contributions in several areas including dynamical systems theory, catastrophe theory, topology, algebraic geometry, classical mechanics and singularity theory, including posing the ADE classification problem, since his first main result—the partial solution of Hilbert's thirteenth problem in 1957 at the age of 19.
A relentless tour-de-force that exposes the landscape of differential equations as part of differential geometry and topology. If you have a solid basis in either of these topics, this text is an eye-opener to applying them directly to solving and seeing the behavior of ODEs with little rote calculation.
I most definitely enjoy reading Arnold. This one is not a book for science and engineering students who want to quickly learn how to solve differential equations, but rather for those who need to know the why and how come.
Excellent geometric insight, but occasionally annoying tendency to write "it is obvious that..." for rather (notationally or logically) tricky parts of proofs. There is a fair number of errors, though these usually do not hamper understanding.
Certain sections of the text may demand unexpectedly high levels of prior knowledge. In addition, there are some diagrams with crucial explanations (or no explanations at all) that are ambiguous, which can be frustrating to interpret.
However, if one sticks to the reading and works through the problems as best they can, they will benefit substantially from it. It should be mentioned that it is often very difficult to treat proofs formally (partly because Arnold himself often does not give entirely formal proofs for the theorems proven). This therefore encourages an appreciation of the geometric view of ODEs.
A coherent, well-formed introduction to ODEs with lots of diagrams. V. I. Arnol'd takes a very visual (and practical) approach, combining concepts from physics with early differential geometry. This is a great book if you want to learn the how and why of differential equations in a less formal setting.