We develop algorithms to compute the differential Galois group $G$ associated to a parameterized second-order homogeneous linear differential equation of the form \[ \tfrac{ \partial^2}{ \partial x^2} Y + r_1 \tfrac{ \partial}{ \partial x} Y + r_0 Y = 0, \] where the coefficients $r_1, r_0 \in F(x)$ are rational functions in $x$ with coefficients in a partial differential field $F$ of characteristic zero. Our work relies on the procedure developed by Dreyfus to compute $G$ under the assumption that $r_1 = 0$. We show how to complete this procedure to cover the cases where $r_1 \neq 0$, by reinterpreting a classical change of variables procedure in Galois-theoretic terms.
Categories: math.AC math.CA