\[ \boxed { {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) =-2\,{\frac {x{\frac {\rm d}{{\rm d}x}}y \left ( x \right ) }{{x}^{2}-1}}-{\frac {v \left ( v+1 \right ) y \left ( x \right ) }{{x}^{2} \left ( {x}^{2}-1 \right ) }}=0} \]
Mathematica: cpu = 0.101513 (sec), leaf count = 86 \[ \left \{\left \{y(x)\to c_1 i^{-v} x^{-v} \, _2F_1\left (\frac {1}{2}-\frac {v}{2},-\frac {v}{2};\frac {1}{2}-v;x^2\right )+c_2 i^{v+1} x^{v+1} \, _2F_1\left (\frac {v}{2}+\frac {1}{2},\frac {v}{2}+1;v+\frac {3}{2};x^2\right )\right \}\right \} \]
Maple: cpu = 0.063 (sec), leaf count = 57 \[ \left \{ y \left ( x \right ) ={\it \_C1}\, {\mbox {$_2$F$_1$}(-{\frac {v}{2}},{\frac {1}{2}}-{\frac {v}{2}};\,{\frac {1}{2}}-v;\,{x}^{2})} {x}^{-v}+{\it \_C2}\, {\mbox {$_2$F$_1$}(1+{\frac {v}{2}},{\frac {1}{2}}+{\frac {v}{2}};\,{\frac {3}{2}}+v;\,{x}^{2})} {x}^{v+1} \right \} \]