\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =2\,{\frac {x \left ( 2\,a-1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{{x}^{2}-1}}-{\frac { \left ( {x}^{2} \left ( 2\,a \left ( 2\,a-1 \right ) -v \left ( v+1 \right ) \right ) +2\,a+v \left ( v+1 \right ) \right ) y \left ( x \right ) }{ \left ( {x}^{2}-1 \right ) ^{2}}}=0} \]
Mathematica: cpu = 0.036005 (sec), leaf count = 32 \[ \left \{\left \{y(x)\to c_1 \left (x^2-1\right )^a P_v(x)+c_2 \left (x^2-1\right )^a Q_v(x)\right \}\right \} \]
Maple: cpu = 0.032 (sec), leaf count = 29 \[ \left \{ y \left ( x \right ) ={\it \_C1}\, \left ( {x}^{2}-1 \right ) ^{a }{\it LegendreP} \left ( v,x \right ) +{\it \_C2}\, \left ( {x}^{2}-1 \right ) ^{a}{\it LegendreQ} \left ( v,x \right ) \right \} \]