First off, I'm very sorry if this sort of question is not allowed here. I've seen a couple similar questions on the OverFlow site, but I think discussions of basic material should be kept to this site.
Anyway, in my ODE course we are using the introductory ODE text written by Jack Hale: http://www.amazon.com/Ordinary-Differential-Equations-Dover-Mathematics/dp/0486472116/ref=sr_1_1?ie=UTF8&qid=1350512056&sr=8-1&keywords=jack+Hale
Actually, it's only introductory in the sense that it is self-contained, but it is rather advanced (think of those Dover reprints in size 8 font, half of which is written in Greek letters). The table of contents and first pages give a hint of the level and material covered (see below).
The proofs encountered in ODE seem to me VERY VERY unintuitive, in the sense that one (without experience) would be very unlikely to ever reproduce such proofs on their own. Often they begin by establishing (without any motivation) estimates which are then used to prove things like "now we see such and such is a nondecreasing," and only once three fourths through the proof does one see why it was even necessary to establish any of the prerequisite facts.
I'm hoping this is just the style of the author, but perhaps this is just the specific taste of ODE theory. In any case, I would like some recommendations for texts that cover similar material on ODE theory that people here have found useful in the past. There are many ODE texts, and they cover different parts of ODE theory. This text (and our course) is aimed specifically at establishing the general theoretical framework for ODE theory (e.g. preliminaries on fixed point theorems/Banach spaces, Peano existence, Picard uniqueness, continuation of solutions, continuous dependence of parameters, differential estimates, further theory on linear systems, etc.), and then going straight to stability analysis (e.g. analysis of linear systems, perturbations of non-linear systems, Poincare-Bendixson theory, and Liapunov methods) and finally perturbation methods (e.g. asymptotic expansions, averaging, multiple scales, etc.). In other words, this is not a course on elementary solution methods encountered in undergraduate courses, nor is it a course on advanced analytic topics such as Sturm-Louiville theory and eigenfunction expansions. It is very much an "applied" course.
Thank you in advance, and again I apologize if this is not strictly a "Math Stackexchange" question.
EDIT (1):
I know that several people have used Strogatz' non-linear dynamics text, which covers the ladder two topics discussed (actually, it covers very little on perturbation methods). However, this text is extremely non-rigorous, and almost nothing is proved (it has the flavor of a catalog of various methods and corresponding examples). So it is not the companion text I am looking for.
Best Answer
I gave a similar answer on MathOverflow some time ago. No ODE textbook covers everything. There are very few, if perhaps none at all, that cover simultaneously the general theory in the sense of analysis, such as Sturm-Liouville theory and considerations of linear differential operators, and which also covers ODE from a dynamical system/geometry point of view.
For the former viewpoint, Theory of Ordinary Differential Equations by Coddington is considered a classic. It's slightly outdated but it covers all the theory. For the latter viewpoint, there are lots of different texts, but I've used Differential Equations and Dynamical Systems by Lawrence Perko. Both are fully rigorous and have fairly lucid expositions. However, I must admit that both are a little bit dry.
You did mention, however, that you're looking for intuition, and while you protest that Strogatz's book is not very rigorous, Strogatz is a fantastic writer and his explanations are very good at developing intuition, in my opinion. I would not try to look too hard for something that covers all the bases.