Kovacic's algorithm

Summary

Kovacic's algorithm decides whether a second-order linear differential equation with rational function coefficients has a Liouvillian solution, and finds one when it exists. It works on the reduced form $y'' = r(x)\, y$ with $r \in \C(x)$. It is based on the classification of the algebraic subgroups of $\mathrm{SL}_2(\C)$, the possible differential Galois groups of such equations. It is implemented in computer algebra systems and is the standard tool for closed-form solution of these equations. See the definition.