\[ y''(x)=-\frac {y(x) \left (-n^2-v (v+1) x^2\right )}{x^2 \left (x^2+1\right )}-\frac {\left (2 x^2+1\right ) y'(x)}{x \left (x^2+1\right )} \] ✓ Mathematica : cpu = 0.205962 (sec), leaf count = 90
\[\left \{\left \{y(x)\to c_1 x^{-n} \, _2F_1\left (-\frac {n}{2}-\frac {v}{2},-\frac {n}{2}+\frac {v}{2}+\frac {1}{2};1-n;-x^2\right )+c_2 x^n \, _2F_1\left (\frac {n}{2}-\frac {v}{2},\frac {n}{2}+\frac {v}{2}+\frac {1}{2};n+1;-x^2\right )\right \}\right \}\] ✓ Maple : cpu = 0.063 (sec), leaf count = 29
\[y \relax (x ) = \LegendreQ \left (v , n , \sqrt {x^{2}+1}\right ) c_{2}+\LegendreP \left (v , n , \sqrt {x^{2}+1}\right ) c_{1}\]