I am TA-ing a introductory course on ODEs and PDEs this year. At my university most introductory math courses can be taken at "basic" and "extended" levels. This one is the extended one.
My students seem to be not engaged in the subject. I don't blame them. They have already seen some abstract mathematical theories -- general topology, measure theory, abstract algebra. What I am teaching them is a lot of methods for solving specific ODEs. There are of course parts of general theory (existence theorems and some qualitative theory) and exercises to prove something, but mostly its calculation.
They long for some abstraction, general framework. Unfortunately, i don't know how to provide it to them without functional analysis (which they haven't had).
Please, give me any of your thoughts and advice on this problem.