\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-1/2\,{\frac { \left ( 3\,x-1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{x \left ( x-1 \right ) }}-1/4\,{\frac { \left ( -v \left ( v+1 \right ) \left ( x-1 \right ) ^{2}-4\,{n}^{2}x \right ) y \left ( x \right ) }{{x}^{2} \left ( x-1 \right ) ^{2}}}=0} \]
Mathematica: cpu = 0.378548 (sec), leaf count = 217 \[ \left \{\left \{y(x)\to c_2 (-1)^{\frac {1}{2} (-2 v-3)+1} x^{\frac {1}{4} (-2 v-3)+1} e^{\frac {1}{4} (-2 \log (1-x)-\log (x))} (x-1)^{\frac {1}{2} \left (n+\frac {1}{2} (2 n+1)+\frac {1}{2} (-2 v-3)+v+2\right )} \, _2F_1\left (\frac {1}{2} (2 n+1)+\frac {1}{2} (-2 v-3)+1,n+\frac {1}{2} (-2 v-3)+v+2;\frac {1}{2} (-2 v-3)+2;x\right )+c_1 x^{\frac {1}{4} (2 v+3)} e^{\frac {1}{4} (-2 \log (1-x)-\log (x))} (x-1)^{\frac {1}{2} \left (n+\frac {1}{2} (2 n+1)+\frac {1}{2} (-2 v-3)+v+2\right )} \, _2F_1\left (\frac {1}{2} (2 n+1),n+v+1;\frac {1}{2} (2 v+3);x\right )\right \}\right \} \]
Maple: cpu = 0.047 (sec), leaf count = 74 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{-{\frac {v}{2}}} \left ( x-1 \right ) ^{-n} {\mbox {$_2$F$_1$}(-v-n,-n+{\frac {1}{2}};\,{\frac {1}{2}}-v;\,x)}+{ \it \_C2}\,{x}^{{\frac {1}{2}}+{\frac {v}{2}}} \left ( x-1 \right ) ^{-n }{\mbox {$_2$F$_1$}(v-n+1,-n+{\frac {1}{2}};\,{\frac {3}{2}}+v;\,x)} \right \} \]