\[ y''(x)=\frac {2 (2 a-1) x y'(x)}{x^2-1}-\frac {y(x) \left (x^2 (2 a (2 a-1)-v (v+1))+2 a+v (v+1)\right )}{\left (x^2-1\right )^2} \] ✓ Mathematica : cpu = 0.0288032 (sec), leaf count = 32
DSolve[Derivative[2][y][x] == -(((2*a + v*(1 + v) + (2*a*(-1 + 2*a) - v*(1 + v))*x^2)*y[x])/(-1 + x^2)^2) + (2*(-1 + 2*a)*x*Derivative[1][y][x])/(-1 + x^2),y[x],x]
\[\left \{\left \{y(x)\to c_1 \left (x^2-1\right )^a P_v(x)+c_2 \left (x^2-1\right )^a Q_v(x)\right \}\right \}\] ✓ Maple : cpu = 0.051 (sec), leaf count = 23
dsolve(diff(diff(y(x),x),x) = 2*x*(2*a-1)/(x^2-1)*diff(y(x),x)-(x^2*(2*a*(2*a-1)-v*(v+1))+2*a+v*(v+1))/(x^2-1)^2*y(x),y(x))
\[y \left (x \right ) = \left (x^{2}-1\right )^{a} \left (c_{1} \LegendreP \left (v , x\right )+c_{2} \LegendreQ \left (v , x\right )\right )\]