\[ y''(x)=\frac {v (v+1) y(x)}{x^2}-\frac {2 x y'(x)}{x^2-1} \] ✓ Mathematica : cpu = 0.0736504 (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 = 0.631 (sec), leaf count = 47
\[ \left \{ y \left ( x \right ) ={\it \_C1}\,{\mbox {$_2$F$_1$}({\frac {1}{2}},-v;\,{\frac {1}{2}}-v;\,{x}^{2})}{x}^{-v}+{\it \_C2}\,{\mbox {$_2$F$_1$}({\frac {1}{2}},v+1;\,{\frac {3}{2}}+v;\,{x}^{2})}{x}^{v+1} \right \} \]