\[ y''(x)=\frac {a^2 y(x)}{\left (x^2-1\right )^2}-\frac {2 x y'(x)}{x^2-1} \] ✓ Mathematica : cpu = 0.0196346 (sec), leaf count = 53
DSolve[Derivative[2][y][x] == (a^2*y[x])/(-1 + x^2)^2 - (2*x*Derivative[1][y][x])/(-1 + x^2),y[x],x]
\[\left \{\left \{y(x)\to c_1 \cosh \left (\frac {1}{2} a (\log (1-x)-\log (x+1))\right )+i c_2 \sinh \left (\frac {1}{2} a (\log (1-x)-\log (x+1))\right )\right \}\right \}\] ✓ Maple : cpu = 0.01 (sec), leaf count = 19
dsolve(diff(diff(y(x),x),x) = -2*x/(x^2-1)*diff(y(x),x)+a^2/(x^2-1)^2*y(x),y(x))
\[y \left (x \right ) = c_{1} \sinh \left (a \arctanh \left (x \right )\right )+c_{2} \cosh \left (a \arctanh \left (x \right )\right )\]