\[ y''(x)=-\frac {b^2 y(x)}{\left (x^2-a^2\right )^2} \] ✓ Mathematica : cpu = 0.270402 (sec), leaf count = 149
\[\left \{\left \{y(x)\to c_1 (x-a)^{\frac {1}{2} \sqrt {1-\frac {b^2}{a^2}}+\frac {1}{2}} (a+x)^{\frac {1}{2}-\frac {1}{2} \sqrt {1-\frac {b^2}{a^2}}}-\frac {c_2 (x-a)^{\frac {1}{2}-\frac {1}{2} \sqrt {\frac {a^2-b^2}{a^2}}} (a+x)^{\frac {1}{2} \sqrt {\frac {a^2-b^2}{a^2}}+\frac {1}{2}}}{2 a \sqrt {\frac {a^2-b^2}{a^2}}}\right \}\right \}\] ✓ Maple : cpu = 0.118 (sec), leaf count = 77
\[\left \{y \left (x \right ) = \sqrt {\left (a -x \right ) \left (a +x \right )}\, \left (c_{1} \left (\frac {a -x}{a +x}\right )^{\frac {\sqrt {a^{2}-b^{2}}}{2 a}}+c_{2} \left (\frac {a -x}{a +x}\right )^{-\frac {\sqrt {a^{2}-b^{2}}}{2 a}}\right )\right \}\]