Can differential equations be classed into their own complexity classes?
A first observation in the direction of decidability of ODEs is this paper by Avigad, Clarke and Gao, which classifies the complexity of $\delta$-decidability, in which solutions are to be found within a certain bounded error (the "delta") in one direction.

one of the main results is that $\delta$-solvability of (Lipschitz-continuous) ODEs is $\mathrm{PSPACE}$-complete.