\[ y''(x)=-\frac {y(x) \left (-a^2-\lambda \left (x^2-1\right )\right )}{\left (x^2-1\right )^2}-\frac {2 x y'(x)}{x^2-1} \] ✓ Mathematica : cpu = 0.0149352 (sec), leaf count = 48
DSolve[Derivative[2][y][x] == -(((-a^2 - lambda*(-1 + x^2))*y[x])/(-1 + x^2)^2) - (2*x*Derivative[1][y][x])/(-1 + x^2),y[x],x]
\[\left \{\left \{y(x)\to c_1 P_{\frac {1}{2} \left (\sqrt {4 \lambda +1}-1\right )}^a(x)+c_2 Q_{\frac {1}{2} \left (\sqrt {4 \lambda +1}-1\right )}^a(x)\right \}\right \}\] ✓ Maple : cpu = 0.053 (sec), leaf count = 37
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)-(-a^2-lambda*(x^2-1))/(x^2-1)^2*y(x),y(x))
\[y \left (x \right ) = c_{1} \LegendreP \left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right )+c_{2} \LegendreQ \left (\frac {\sqrt {1+4 \lambda }}{2}-\frac {1}{2}, a , x\right )\]