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

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

y - y' - y'' - y''' - ... = 0
Every (differentiable) function satisfies many differential equations. Shouldn't be too hard too cook up an "ugly" equatipn satisfied by, say, e^x.
IDK if this qualifies but the Lindblad equation is pretty ugly when you write it all the way out.
A permitted solution to general relativity field equations is empty space, so most terms are zero, although there is still inertia.
take any nice equation of two variables and convert it from polar form into cartesian coordinates.

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