\[ y''(x)=-\frac {v (v+1) y(x)}{x^2 \left (x^2-1\right )}-\frac {2 x y'(x)}{x^2-1} \] ✓ Mathematica : cpu = 0.082079 (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 = 0.646 (sec), leaf count = 57
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}(-{\frac {v}{2}},{\frac {1}{2}}-{\frac {v}{2}};\,{\frac {1}{2}}-v;\,{x}^{2})}{x}^{-v}+{\it \_C2}\,{\mbox {$_2$F$_1$}(1+{\frac {v}{2}},{\frac {1}{2}}+{\frac {v}{2}};\,{\frac {3}{2}}+v;\,{x}^{2})}{x}^{v+1} \right \} \]