Examples of analytically unsolvable ODEs in classical mechanics

[dipole]

Hi, I'm trying to come up with some examples of ODEs (not PDES) in physics that are unsolvable analytically, and for some reason I'm drawing a blank.

There are a few obvious examples from classical mechanics such as a pendulum and coupled springs etc... but beyond that I can't seem to recall coming across any.

Can anyone think of some ODEs that describe interesting physics and whose derivation can be understood without too much background knowledge?

 

[SteamKing]

The queen of unsolvable ODEs IMO is the Three-Body Problem. Solutions have been obtained for certain situations, but a general solution has yet to be obtained.

See: http://www.scholarpedia.org/article/Three_body_problem