That is really, really hard to believe. I can't say that I remember any discussion of applications in my differential equations course (also at a Big Ten school), but I'm positive the professor could have provided them.

I didn't have a particular need for examples in a course, as differential equations were held up as the holy grail of math by my father, an optical engineer -- he used them at work fairly often, and frequently they were what made him better able to solve a problem than an engineer who wasn't comfortable using them.