I agree (differential geometry as well). Ironically, though, implicit differential equations tend to melt my brain, or rather modern approaches to them melt my

I agree (differential geometry as well). Ironically, though, implicit differential equations tend to melt my brain, or rather modern approaches to them melt my brain (contact form on the first jet bundle anyone?) and classic ones look sensible at any single step but utterly magical on the whole (what the hell is a discriminant set? why do we have exceptional solutions?).