I'm trying to explain to a friend the power of differential equations in modelling math stuff. Does anyone have some really engaging/"juicy" examples? I looked at this question and found some good ones, for example:
A smooth football having the shape of a prolate spheroid 12 inches long and 6 inches thick is lying outdoors in a rainstorm. Find the paths along which the water will run down its sides.
Discovering a planet should be a good example. To develop a bit: the orbits of planets are predicted by differential equations (Newton's laws). The observed movement of planet Uranus did not exactly fit the predictions (there was a very small discrepancy). A mathematician (well, two, actually), took this on and put the following hypothesis: the planet was perturbed by a nearby, yet-unknown, planet. He did lots of computations, solved more differential equations, and emerged with the possible position of the unknown planet. He mailed a friend at an observatory with the said position. There was indeed a new planet at this place.
