\[ \frac {n x Q_n(x)-n Q_{n-1}(x)}{x^2-1}-n (n+1) y(x)+\left (x^2-1\right ) y''(x)=0 \] ✗ Mathematica : cpu = 299.999 (sec), leaf count = 0 , timed out
$Aborted
✓ Maple : cpu = 0.154 (sec), leaf count = 409
\[ \left \{ y \left ( x \right ) =3\, \left ( 1+x \right ) \left ( -{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})} \left ( n+1 \right ) \int \!1/3\,{\frac {x{\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})} \left ( x{\it LegendreQ} \left ( n,x \right ) -{\it LegendreQ} \left ( n+1,x \right ) \right ) }{ \left ( 1+x \right ) ^{3} \left ( \left ( {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}+ \left ( {n}^{2}+n-2 \right ) {x}^{2}{\mbox {$_2$F$_1$}(n/2+2,3/2-n/2;\,3/2;\,{x}^{2})} \right ) {\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})}-1/3\,{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}{\mbox {$_2$F$_1$}(-n/2+2,n/2+5/2;\,5/2;\,{x}^{2})}{x}^{2} \left ( n+3 \right ) \left ( n-2 \right ) \right ) \left ( x-1 \right ) ^{3}}}\,{\rm d}x+x{\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})} \left ( n+1 \right ) \int \!1/3\,{\frac {{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})} \left ( x{\it LegendreQ} \left ( n,x \right ) -{\it LegendreQ} \left ( n+1,x \right ) \right ) }{ \left ( 1+x \right ) ^{3} \left ( \left ( {\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}+ \left ( {n}^{2}+n-2 \right ) {x}^{2}{\mbox {$_2$F$_1$}(n/2+2,3/2-n/2;\,3/2;\,{x}^{2})} \right ) {\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})}-1/3\,{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}{\mbox {$_2$F$_1$}(-n/2+2,n/2+5/2;\,5/2;\,{x}^{2})}{x}^{2} \left ( n+3 \right ) \left ( n-2 \right ) \right ) \left ( x-1 \right ) ^{3}}}\,{\rm d}x-1/3\,{\mbox {$_2$F$_1$}(-n/2+1,n/2+3/2;\,3/2;\,{x}^{2})}{\it \_C1}\,x-1/3\,{\mbox {$_2$F$_1$}(n/2+1,-n/2+1/2;\,1/2;\,{x}^{2})}{\it \_C2} \right ) \left ( x-1 \right ) \right \} \]