\[ y''(x)=-\frac {a y(x)}{\left (x^2-1\right )^2} \] ✓ Mathematica : cpu = 0.11485 (sec), leaf count = 110
DSolve[Derivative[2][y][x] == -((a*y[x])/(-1 + x^2)^2),y[x],x]
\[\left \{\left \{y(x)\to \frac {c_2 (x+1)^{\frac {\sqrt {1-a}}{2}+\frac {1}{2}} (1-x)^{\frac {1}{2}-\frac {\sqrt {1-a}}{2}}}{2 \sqrt {1-a}}+c_1 (x+1)^{\frac {1}{2}-\frac {\sqrt {1-a}}{2}} (1-x)^{\frac {1}{2} \left (\sqrt {1-a}+1\right )}\right \}\right \}\] ✓ Maple : cpu = 0.056 (sec), leaf count = 55
dsolve(diff(diff(y(x),x),x) = -a/(x^2-1)^2*y(x),y(x))
\[y \left (x \right ) = \sqrt {x^{2}-1}\, \left (\left (\frac {x -1}{1+x}\right )^{-\frac {\sqrt {1-a}}{2}} c_{2}+\left (\frac {x -1}{1+x}\right )^{\frac {\sqrt {1-a}}{2}} c_{1}\right )\]