\[ \boxed { x \left ( {x}^{2}+1 \right ) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}y \left ( x \right ) - \left ( 2\, \left ( n-1 \right ) {x}^{2}+2\,n-1 \right ) {\frac {\rm d}{{\rm d}x}}y \left ( x \right ) + \left ( v+n \right ) \left ( -v+n-1 \right ) xy \left ( x \right ) =0} \]
Mathematica: cpu = 0.185024 (sec), leaf count = 87 \[ \left \{\left \{y(x)\to c_1 \, _2F_1\left (-\frac {n}{2}-\frac {v}{2},-\frac {n}{2}+\frac {v}{2}+\frac {1}{2};1-n;-x^2\right )+c_2 x^{2 n} \, _2F_1\left (\frac {n}{2}-\frac {v}{2},\frac {n}{2}+\frac {v}{2}+\frac {1}{2};n+1;-x^2\right )\right \}\right \} \]
Maple: cpu = 0.062 (sec), leaf count = 35 \[ \left \{ y \left ( x \right ) ={\it \_C1}\,{x}^{n}{\it LegendreP} \left ( v,n,\sqrt {{x}^{2}+1} \right ) +{\it \_C2}\,{x}^{n}{\it LegendreQ} \left ( v,n,\sqrt {{x}^{2}+1} \right ) \right \} \]