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We propose a new method to compute the unipotent radical $R_u(H)$ of thedifferential Galois group $H$ associated to a parameterized second-orderhomogeneous linear differential equation of the form\[\tfrac{\partial^2}{\partial x^2}Y-qY=0,\] where $q \in F(x)$ is a rationalfunction in $x$ with coefficients in a $\Pi$-field $F$ of characteristic zero,and $\Pi$ is a commuting set of parametric derivations. The procedure developedby Dreyfus reduces the computation of $R_u(H)$ to solving a creativetelescoping problem, whose effective solution requires the assumption that themaximal reductive quotient $H / R_u(H)$ is a $\Pi$-constant linear differentialalgebraic group. When this condition is not satisfied, we compute a new set ofparametric derivations $\Pi'$ such that the associated differential Galoisgroup $H'$ has the property that $H'/ R_u(H')$ is $\Pi'$-constant, and suchthat $R_u(H)$ is defined by the same differential equations as $R_u(H')$. Thusthe computation of $R_u(H)$ is reduced to the effective computation of$R_u(H')$. We expect that an elaboration of this method will be successful inextending the applicability of some recent algorithms developed by Minchenko,Ovchinnikov, and Singer to compute unipotent radicals for higher orderequations.

Computation of the unipotent radical of the differential Galois group for a parameterized second-order linear differential equation