ODE No. 1240

\[ -v (v+1) y(x)+\left (x^2-1\right ) y''(x)+2 x y'(x)=0 \]

✓ Mathematica : cpu = 0.0098459 (sec), leaf count = 18

DSolve[-(v*(1 + v)*y[x]) + 2*x*Derivative[1][y][x] + (-1 + x^2)*Derivative[2][y][x] == 0,y[x],x]

\[\{\{y(x)\to c_1 P_v(x)+c_2 Q_v(x)\}\}\]

✓ Maple : cpu = 0.059 (sec), leaf count = 15

dsolve((x^2-1)*diff(diff(y(x),x),x)+2*x*diff(y(x),x)-v*(v+1)*y(x)=0,y(x))

\[y \left (x \right ) = c_{1} \operatorname {LegendreP}\left (v , x\right )+c_{2} \operatorname {LegendreQ}\left (v , x\right )\]