\[ (n-v-1) (n+v) y(x)-2 (n-1) x y'(x)+\left (x^2-1\right ) y''(x)=0 \] ✓ Mathematica : cpu = 0.0176256 (sec), leaf count = 42
DSolve[(-1 + n - v)*(n + v)*y[x] - 2*(-1 + n)*x*Derivative[1][y][x] + (-1 + x^2)*Derivative[2][y][x] == 0,y[x],x]
\[\left \{\left \{y(x)\to c_1 \left (x^2-1\right )^{n/2} P_v^n(x)+c_2 \left (x^2-1\right )^{n/2} Q_v^n(x)\right \}\right \}\] ✓ Maple : cpu = 0.049 (sec), leaf count = 27
dsolve((x^2-1)*diff(diff(y(x),x),x)-2*(n-1)*x*diff(y(x),x)-(v-n+1)*(v+n)*y(x)=0,y(x))
\[y \left (x \right ) = \left (x^{2}-1\right )^{\frac {n}{2}} \left (\LegendreQ \left (v , n , x\right ) c_{2}+\LegendreP \left (v , n , x\right ) c_{1}\right )\]