Computing the differential Galois group of a one-parameter family of second order linear differential equations

We develop algorithms to compute the differential Galois group corresponding to a one-parameter family of second order homogeneous ordinary linear differential equations with rational function coefficients. More precisely, we consider equations of the form \frac{\partial^2Y}{\partial x^2}+ r_1\frac{\partial Y}{\partial x} +r_2Y=0, where $r_1,r_2\in C(x,t)$ and $C$ is an algebraically closed field of characteristic zero. We work in the setting of parameterized Picard-Vessiot theory, which attaches a linear differential algebraic group to such an equation, that is, a group of invertible matrices whose entries satisfy a system of polynomial differential equations, with respect to the derivation in the parameter-space. We will compute the $\frac{\partial}{\partial t}$-differential-polynomial equations that define the corresponding parameterized Picard-Vessiot group as a differential algebraic subgroup of $\mathrm{GL}_2$.