At a Glance - In the Real World
Differential equations, believe it or not, show up everywhere.
There are math departments studying how differential equations relate to biology papers using differential equations for the study of diseases, books relating differential equations and economics and whole journals devoted to differential equations and their applications. Calculus classes usually introduce differential equations, but don't do a whole lot with them. To get more, there are usually differential-equations-only classes offered after calculus.