This question might have an obvious answer but it eludes me. It is equivalent to asking whether every ODE can be solved in quadratures. For example, a first order ODE can be reduced to quadratures if we can solve for the integration factor $ mu$. This requires solving a first order partial differential equation, and as far as I know there are no general existence theorems on these.
A possible equivalent question would be, given a function $x=f(t,c)$, can we find two functions $F(x,t)$ and $G(c)$, such that $F(f(t,c),t)=G(c)$?