\[ y''(x)=-\frac {y(x) \left (v (v+1) (x-1)-a^2 x\right )}{4 (x-1)^2 x^2}-\frac {(3 x-1) y'(x)}{2 (x-1) x} \] ✓ Mathematica : cpu = 0.307618 (sec), leaf count = 109
\[\left \{\left \{y(x)\to \frac {(-1)^{-v} (x-1)^{\frac {a+1}{2}} x^{-v/2} \left (c_1 (-1)^v x^{v+\frac {1}{2}} \, _2F_1\left (\frac {1}{2} (a+v+1),\frac {1}{2} (a+v+2);v+\frac {3}{2};x\right )-i c_2 \, _2F_1\left (\frac {a-v}{2},\frac {1}{2} (a-v+1);\frac {1}{2}-v;x\right )\right )}{\sqrt {1-x}}\right \}\right \}\]
✓ Maple : cpu = 0.083 (sec), leaf count = 76
\[ \left \{ y \left ( x \right ) = \left ( x-1 \right ) ^{-{\frac {a}{2}}} \left ( {x}^{-{\frac {v}{2}}}{\mbox {$_2$F$_1$}(-{\frac {v}{2}}-{\frac {a}{2}},{\frac {1}{2}}-{\frac {v}{2}}-{\frac {a}{2}};\,{\frac {1}{2}}-v;\,x)}{\it \_C1}+{x}^{{\frac {1}{2}}+{\frac {v}{2}}}{\mbox {$_2$F$_1$}(1+{\frac {v}{2}}-{\frac {a}{2}},{\frac {1}{2}}+{\frac {v}{2}}-{\frac {a}{2}};\,{\frac {3}{2}}+v;\,x)}{\it \_C2} \right ) \right \} \]