Graduate ODE textbook Suppose that a hypothetical math grad student was pretty comfortable with first-year real variables and algebra, and had even studied some other things (algebraic geometry, Riemannian geometry, complex analysis, algebraic topology, algebraic number theory), but had miraculously never taken a differential equations course (despite the geometry) and wanted to learn some ODEs.  What book would you recommend?  This student would be happy to learn more analysis if necessary to understand what's in this ODE book.
In other words: I'm asking for your recommendations for a ODE book that is allowed to have arbitrary prerequisites from analysis and algebra and topology and even geometry, but with no knowledge of differential equations presumed.
Thank you!
(Note: It could have occurred to the hypothetical student to talk to his/her advisor/other faculty members, but in that case the student would still be interested in MathOverflow's response.)
 A: There are way too many approaches to ODEs to have any one book cover them all. I occasionally use a book called Differential Equations and Dynamical Systems, by Lawrence Perko. The focus of this book is on qualitative behavior - existence of fixed points, limit cycles, blow-up solutions, etc.
I would not call this a standard introduction to ODE - it does not cover some of the absolute basics. However, I think the emphasis of this text on geometry, and on using some more modern results, makes the book a decent choice.
Some flaws: The book really only presupposes mastery of analysis. There are some tools missing, in particular from geometry/topology, that could make the presentation a bit cleaner. It sounds like you have a strong geometry/topology background, so maybe this disqualifies this text for you.
For a more classical treatment of ODEs, in particular the treatment of ODEs as linear operators (Sturm-Liouville theory), I might go for Coddington's Theory of Ordinary Differential Equations. It is very classical, but it really does cover all the essential theory. 
A: I really like Ordinary Differential Equations by Jack K. Hale. It's very rigorous
and thorough in the fundamentals, has a great section on periodic linear systems, and covers
some advanced stuff such as integral manifolds.  Arnold, Abraham and Marsden, and Hirsch, Smale and Devaney are also nice, though the emphasis is different.
A: Ordinary Differential Equations by Philip Hartman, John Wiley & sons, 1964 might be what you are looking for.
A: Here's a nice book by Gerald Teschl.
http://www.mat.univie.ac.at/~gerald/ftp/book-ode/ode.pdf
A: My advice is different.  Work through a standard undergrad text on DiffyQs first.  Not one with all the fancy connections to other fields of math that you know.  But one emphasizing manipulation and problem solving and applications. Then after that, go grab some fancy book with all the grad school emphasis on proofs and Sobolev spaces and the like.  
Because 90%+ of those books assume exposure to diffyQs first (in the way that "real analysis" typically assumes "calculus" exposure first).  And it's not just about how books are constructed and how people typically learn.  It's actually more efficient and you will learn more and deeper by learning the content first in terms of problem solving manipulation and later in terms of all the fancy stuff.  It's pedagogically advantageous.  The human brain is not a computer, it learns from imitation and repetition.  This is why you can't teach a young gymnast a double back when they start.  
A good, cheap book for self study is Tenanbaum and Pollard.  800 pages, all problems have answers.  Covers the whole playing field.  And even has a lot of non rigorous proofs.  
P.s.  I'm curious if you have a gap in PDEs also.  Can't see how you can work with PDEs if you don't have undergrad familiarity with ODEs as many problems are solved by converting a PDE to an ODE.
A: Probably not quite right but a GTM book you might be interested in is Olver's book "Applications of Lie Groups to Differential Equations" given your background.
A: Henri Cartan's course in differential calculus does cover quite a few useful things for differential equations, from a high-level point of view : you'll find the notion of differentiation in a generic form, the big theorems are proven (local inversion, Cauchy-Lipschitz, ...).
For the low-level and the explicit, Arnold's "Ordinary differential equations" is a must-read, as Qiaochu Yuan already pointed out.
A: Teschl's book is a good, modern text on ODEs and is available from the AMS in their GSM series. It blends ODEs and dynamical systems. Bob Devaney updated the classic Hirsch and Smale book some years ago. It has an emphasis on dynamical systems. A real classic is the book by Ince available from Dover. It predates the modern methods, but is a very serious book. 
A: If you happen to like Arnold's ODE book mentioned by Julien Puydt it could also be a good idea to look at his more advanced book Geometrical methods in the theory of ordinary differential equations. 
