  We present two algorithms for computing hypergeometric solutions of second
order linear differential operators with rational function coefficients. Our
first algorithm searches for solutions of the form \[ \exp(\int r \,
dx)\cdot{_{2}F_1}(a_1,a_2;b_1;f) \] where $r,f \in \overline{\mathbb{Q}(x)}$,
and $a_1,a_2,b_1 \in \mathbb{Q}$. It uses modular reduction and Hensel lifting.
Our second algorithm tries to find solutions in the form \[ \exp(\int r \,
dx)\cdot \left( r_0 \cdot{_{2}F_1}(a_1,a_2;b_1;f) + r_1
\cdot{_{2}F_1}'(a_1,a_2;b_1;f) \right) \] where $r_0, r_1 \in
\overline{\mathbb{Q}(x)}$, as follows: It tries to transform the input equation
to another equation with solutions of the first type, and then uses the first
algorithm.
