\[ y''(x)=-\frac {b^2 y(x)}{\left (x^2-a^2\right )^2} \] ✓ Mathematica : cpu = 0.194675 (sec), leaf count = 149
DSolve[Derivative[2][y][x] == -((b^2*y[x])/(-a^2 + x^2)^2),y[x],x]
\[\left \{\left \{y(x)\to c_1 (x-a)^{\frac {1}{2} \sqrt {1-\frac {b^2}{a^2}}+\frac {1}{2}} (a+x)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {b^2}{a^2}}}-\frac {c_2 (x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {\frac {a^2-b^2}{a^2}}} (a+x)^{\frac {1}{2} \sqrt {\frac {a^2-b^2}{a^2}}+\frac {1}{2}}}{2 a \sqrt {\frac {a^2-b^2}{a^2}}}\right \}\right \}\] ✓ Maple : cpu = 0.102 (sec), leaf count = 77
dsolve(diff(diff(y(x),x),x) = -b^2/(-a^2+x^2)^2*y(x),y(x))
\[y \left (x \right ) = \sqrt {\left (a -x \right ) \left (x +a \right )}\, \left (\left (\frac {a -x}{x +a}\right )^{-\frac {\sqrt {a^{2}-b^{2}}}{2 a}} c_{2}+\left (\frac {a -x}{x +a}\right )^{\frac {\sqrt {a^{2}-b^{2}}}{2 a}} c_{1}\right )\]