Are there general methods to solve recursive ordinary differential equations? Consider the following two kinds of equations.
Explicit recursion:
Solve for $f_n(x)$ in
$$\frac{\mathrm{d}\left(f_n(x)p(x)\right)}{\mathrm{d}x} = f_{n-1}(x)q(x) + r(x)\\ f_0(x)=h(x)$$
where $p(x), q(x), r(x), h(x)$ and $f_i(x), \forall i \in \mathbb{N}$ are real analytic functions. Assume the equation have solution(s).
Implicit recursion:
Consider the same set of equations, with a more general implicit acyclic recursive relation $ m(n): \mathbb{N} \rightarrow \mathbb{N}$, instead of the explicit recursion $n \rightarrow n-1$ as in the first equation. Namely:
$$\frac{\mathrm{d}\left(f_n(x)p(x)\right)}{\mathrm{d}x} = f_{m(n)}(x)q(x) + r(x)$$
Is there a general method of solving these? If not, is there a proof (apart from the "obvious" intuition that you would need to know the structure of the recursive map to actually solve it, so a general method of solving is a no-no) that such a general method does not exist?