\[ y''(x)=-\frac {a y(x)}{\left (x^2-1\right )^2} \] ✓ Mathematica : cpu = 0.0700843 (sec), leaf count = 106
DSolve[Derivative[2][y][x] == -((a*y[x])/(-1 + x^2)^2),y[x],x]
\[\left \{\left \{y(x)\to \frac {c_2 \sqrt {1-x^2} (x+1)^{\sqrt {1-a}} (1-x)^{-\sqrt {1-a}} e^{-\sqrt {1-a} \tanh ^{-1}(x)}}{2 \sqrt {1-a}}+c_1 \sqrt {1-x^2} e^{-\sqrt {1-a} \tanh ^{-1}(x)}\right \}\right \}\] ✓ Maple : cpu = 0.049 (sec), leaf count = 55
dsolve(diff(diff(y(x),x),x) = -a/(x^2-1)^2*y(x),y(x))
\[y \left (x \right ) = \sqrt {x^{2}-1}\, \left (\left (\frac {x -1}{1+x}\right )^{-\frac {\sqrt {1-a}}{2}} c_{2}+\left (\frac {x -1}{1+x}\right )^{\frac {\sqrt {1-a}}{2}} c_{1}\right )\]