\[ (1-v) v y(x)+\left (x^2+1\right ) y''(x)+2 x y'(x)=0 \] ✓ Mathematica : cpu = 0.0107046 (sec), leaf count = 30
DSolve[(1 - v)*v*y[x] + 2*x*Derivative[1][y][x] + (1 + x^2)*Derivative[2][y][x] == 0,y[x],x]
\[\{\{y(x)\to c_1 P_{v-1}(i x)+c_2 Q_{v-1}(i x)\}\}\] ✓ Maple : cpu = 0.041 (sec), leaf count = 25
dsolve((x^2+1)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-v*(v-1)*y(x)=0,y(x))
\[y \left (x \right ) = c_{1} \LegendreP \left (v -1, i x \right )+c_{2} \LegendreQ \left (v -1, i x \right )\]