Computational complexity of smooth differential equations

The computational complexity of the solution h to the ordinary differential equation h(0) = 0, h'(t) = g(t,h(t)) under various assumptions on the function g has been investigated. Kawamura showed in 2010 that the solution h can be PSPACE-hard even if g is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of g and obtain the following results: the solution h can still be PSPACE-hard if g is assumed to be of class C¹; for each k > 2, the solution h can be hard for the counting hierarchy even if g is of class C^k.