Kovacic's algorithm solves second-order linear homogeneous differential equations with rational function coefficients.
I'm wondering if anybody has extended this algorithm to handle algebraic function coefficients.
More specifically, given $a,b \in {\mathbb C}(x)$, Kovacic's algorithm determines if there exists "closed form" solutions (i.e, finite rational combinations of exponentials, integrals, and roots) of the equation
$$r'' + ar' + br = 0.$$
Here $r$ is not constrained to be in ${\mathbb C}(x)$, and may not exist in a closed form. In the later case, the algorithm's failure to find a closed form solution is a proof that no such solution exists.
My question: if we add an irreducible algebraic curve to the mix, i.e, an irreducible polynomial $f(x,y) \in {\mathbb C}[x,y]$, then consider coefficients $a,b \in {\mathbb C}(x,y)$, is there a known algorithm to solve:
$$r'' + ar' + br = 0$$
with $r$, as before, required to be a closed form expression?
