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

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

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

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