I've been teaching calculus courses for a while now, and something always bothers me each time I teach it. Students always seem to have trouble connecting with the differential equation material for the following reason: I always tell them that they are one of the most important topics for applications of calculus (this is mainly a course for students in the sciences) and that all sorts of fields use them...and then all I have to tell them are things that are to a certain extent quite dull: exponential growth, Newton's law of cooling, the logistic equation, and a few other of the classics. While each of these is quite important and do have broad applications, I've never seen anyone be shocked to learn that populations of rabbits breeding in the wild grow approximately exponentially.

My knowledge of applied fields isn't terrible, but I'm still at a loss as to what plausible models I could teach them about where the global results are not immediately obvious, so I ask: what are some simple differential equations, simple enough for a freshman calculus class, which occur in the sciences and have behavior interesting enough the catch peoples' interest?