\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-1/2\,{\frac { \left ( 3\,x-1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{x \left ( x-1 \right ) }}-1/4\,{\frac { \left ( v \left ( v+1 \right ) \left ( x-1 \right ) -{a}^{2}x \right ) y \left ( x \right ) }{{x}^{2} \left ( x-1 \right ) ^{2}}}=0} \]
Mathematica: cpu = 0.291037 (sec), leaf count = 235 \[ \left \{\left \{y(x)\to c_2 (-1)^{\frac {1}{2} (-2 v-3)+1} x^{\frac {1}{4} (-2 v-3)+1} e^{\frac {1}{4} (-2 \log (1-x)-\log (x))} (x-1)^{\frac {1}{2} \left (\frac {1}{2} (a+v+1)+\frac {1}{2} (a+v+2)+\frac {1}{2} (-2 v-3)+1\right )} \, _2F_1\left (\frac {1}{2} (-2 v-3)+\frac {1}{2} (a+v+1)+1,\frac {1}{2} (-2 v-3)+\frac {1}{2} (a+v+2)+1;\frac {1}{2} (-2 v-3)+2;x\right )+c_1 x^{\frac {1}{4} (2 v+3)} e^{\frac {1}{4} (-2 \log (1-x)-\log (x))} (x-1)^{\frac {1}{2} \left (\frac {1}{2} (a+v+1)+\frac {1}{2} (a+v+2)+\frac {1}{2} (-2 v-3)+1\right )} \, _2F_1\left (\frac {1}{2} (a+v+1),\frac {1}{2} (a+v+2);\frac {1}{2} (2 v+3);x\right )\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 82 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{-{\frac {v}{2}}} \left ( x-1 \right ) ^{-{\frac {a}{2}}} {\mbox {$_2$F$_1$}(-{\frac {v}{2}}-{\frac {a}{2}},{\frac {1}{2}}-{\frac {v}{2}}-{\frac {a}{2}};\,{\frac {1}{2}}-v;\,x)} +{\it \_C2}\,{x}^{{\frac {1}{2}}+{\frac {v}{2}}} \left ( x-1 \right ) ^{ -{\frac {a}{2}}} {\mbox {$_2$F$_1$}(1+{\frac {v}{2}}-{\frac {a}{2}},{\frac {1}{2}}+{\frac {v}{2}}-{\frac {a}{2}};\,{\frac {3}{2}}+v;\,x)} \right \} \]