I took differential equations as a mechanical engineering student, and therefore my learning was very much in the line of "here's what we can solve, here's how we solve it, here's how we approximate what we cannot solve". I have since gotten a Masters in Mathematics, where conversations with my adviser have made it very clear how woeful my diff eq knowledge is from a qualitative perspective.
I would much appreciate a textbook that gives a rigorous introduction to the qualitative study of differential equations. My background in linear algebra and analysis is solid. Any suggestions?