That I could create as a classical mechanics class project? Other than the classical examples that we see in textbooks (catenary, brachistochrone, Fermat, etc..)
Here is one I just made up, but it has a nice flavor— suppose you have a 2-d bullet going very fast through a 2-d gas. The gas molecules reflects specularly off the bullet, making glancing collisions. What shape of bullet of a fixed area has the least drag?
This problem gives
And the equation for y’ you get is
Which you can solve in a series by plugging in y=λx22 and iterating a few times using the relation above as a recursion.