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I have just completed my first year study and know elementary analysis and a little bit functional analysis. I found that most of the ODE books just focus on calculation but no substantial explanation of theorems.Can someone suggest some ODE books which are from a more theoretical point of view?

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Possible duplicate:… – Belgi Jul 27 '12 at 16:37
Its not an ODE book, and it be heavy going at an early stage, but I like the (brief) treatment of ODEs in Kantorovich & Akilov's "Functional Analysis". In particular, it provides a fixed-point scheme (as in Picard) that is useful for showing continuity of solutions with respect to parameters. – copper.hat Jul 27 '12 at 17:30

A classical theoretical book on ODE is Hartman.

A very good book, and slightly less demanding than Hartman is Hale's book

A geometric picture of differential equations is given in two Arnold's books: one and two

ODE from a dynamical system theory point of view are presented in Wiggins' book

Update: Have no idea how, but I read that the question was about a second theoretical ODE course. For the first course in ODE none of the books that I mentioned (except Arnold's one) suits.

The best first theoretical book on ODE is, for my taste, is Hirsch and Smale.

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Hartman is great for references, but I would kill myself if I had to use it for self-study as my first ODE course. Arnold's books make excellent reading. – user31373 Jul 27 '12 at 20:13
@LeonidKovalev Ups. Somehow I understood that OP asked about a second course in differential equations. I update my post. – Artem Jul 29 '12 at 15:05

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