  We propose a new method to compute the unipotent radical $R_u(H)$ of the
differential Galois group $H$ associated to a parameterized second-order
homogeneous linear differential equation of the form
\[\tfrac{\partial^2}{\partial x^2}Y-qY=0,\] where $q \in F(x)$ is a rational
function in $x$ with coefficients in a $\Pi$-field $F$ of characteristic zero,
and $\Pi$ is a commuting set of parametric derivations. The procedure developed
by Dreyfus reduces the computation of $R_u(H)$ to solving a creative
telescoping problem, whose effective solution requires the assumption that the
maximal reductive quotient $H / R_u(H)$ is a $\Pi$-constant linear differential
algebraic group. When this condition is not satisfied, we compute a new set of
parametric derivations $\Pi'$ such that the associated differential Galois
group $H'$ has the property that $H'/ R_u(H')$ is $\Pi'$-constant, and such
that $R_u(H)$ is defined by the same differential equations as $R_u(H')$. Thus
the 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 in
extending the applicability of some recent algorithms developed by Minchenko,
Ovchinnikov, and Singer to compute unipotent radicals for higher order
equations.
