Question

What are some examples of "ugly" differential equations with surprisingly simple solutions?

Answer 1

I'm not sure this quite fits here but I was very surprised by how the KdV- and KP hierachies are "solved by moduli theory". Perhaps these equations are not ugly and perhaps Weierstrass and theta functions are not simple though.

Answer 2

To be honest, even "nice" differential equations can have very ugly solutions, and most equations don't even have solutions, so I wouldn't bet on an ugly equation to have a "nice" solution.
Quite interested in what other responses you might get though

Answer 3

This might not be "ugly" but it's an infinite-order ODE:

y - y' - y'' - y''' - ... = 0

Answer 4

Every (differentiable) function satisfies many differential equations. Shouldn't be too hard too cook up an "ugly" equatipn satisfied by, say, e^x.

Answer 5

IDK if this qualifies but the Lindblad equation is pretty ugly when you write it all the way out.

Answer 6

A permitted solution to general relativity field equations is empty space, so most terms are zero, although there is still inertia.

Answer 7

take any nice equation of two variables and convert it from polar form into cartesian coordinates.