\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-2\,{\frac {x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{{x}^{2}-1}}+{\frac {v \left ( v+1 \right ) y \left ( x \right ) }{{x}^{2}}}=0} \]
Mathematica: cpu = 0.091512 (sec), leaf count = 68 \[ \left \{\left \{y(x)\to c_1 i^{-v} x^{-v} \, _2F_1\left (\frac {1}{2},-v;\frac {1}{2}-v;x^2\right )+c_2 i^{v+1} x^{v+1} \, _2F_1\left (\frac {1}{2},v+1;v+\frac {3}{2};x^2\right )\right \}\right \} \]
Maple: cpu = 0.046 (sec), leaf count = 47 \[ \left \{ y \left ( x \right ) ={\it \_C1}\, {\mbox {$_2$F$_1$}({\frac {1}{2}},-v;\,{\frac {1}{2}}-v;\,{x}^{2})}{x}^ {-v}+{\it \_C2}\, {\mbox {$_2$F$_1$}({\frac {1}{2}},v+1;\,{\frac {3}{2}}+v;\,{x}^{2})}{x} ^{v+1} \right \} \]