Book reference for theory of differential equations (not Coddington's book) I'm looking for references to study theory of ordinary differential equations. I'm looking for a similar book to Coddington's book, theory of ordinary differential equations but not this one, because this is a little old. I've already taken a course of (applied) differential equations but now I want to delve into the theory. I love Coddington's book but it is quite old. Also I like have more than one reference.
Thank you in advance for your time.
 A: The book that I consult always on such matters is the little book of Vladimir Arnol'd [1], written on the basis of the university courses held by the Author at the Moscow University during the years 1968-1969 and 1969-1970. In the book, the theory is fully developed, meaning that all standard existence and uniqueness theorems are proved and the theory is developed from scratch, requiring only basic mathematical analysis and linear algebra as prerequisites. But there is more: Arnol'd develop the theory with a strong geometrical and topological flavor. Essentially he offers a picture of the theory as a part of the geometry and topology of the phase space. Also despite being mathematically rigorous, the text is full of pictures and examples, and also the writing style is very informal: also 


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*It has been translated in to a variety of languages so, whatever is your mother tongue, it is extremely likely that you can find an edition that you can read fluently.

*It has a companion volume (Arnol’d, V. I., Geometrical methods in the theory of ordinary differential equations), which deals with the more advanced aspects of the theory, written in the same style and translated in as many languages.


On the other side, the stylistic choice of the author implies that some topics are at least only touched:


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*It deals only with ODEs in the real domain: power series solutions, special functions and the like are not dealt at all.

*It is not a collection of integration methods: however the author indicates where it is possible to find large collections of exercises (i.e. in the books of Ince and Kamke).


In sum, I think that the book [1] is a wonderful reference on such topic.
[1] Vladimir Igorevic Arnol'd, "Ordinary differential equations", various editions from MIT Press and from Springer-Verlag, MR1162307 Zbl 0744.34001.
