This is an old question, but it was just bumped to the top and I noticed that my favorite ODE book wasn't listed.

Anyway, I highly recommend Hurewicz's beautiful little book "Lectures on ordinary differential equations". It's extremely short, efficient, and easy to read, and it contains everything a non-analyst needs to know about ODE's. It would probably be hard to teach a course from it, but for self-study it is perfect.