\[ y''(x)=-\frac {b^2 y(x)}{\left (x^2-a^2\right )^2} \] ✓ Mathematica : cpu = 0.192145 (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 = 0.219 (sec), leaf count = 77
\[ \left \{ y \left ( x \right ) =\sqrt { \left ( a-x \right ) \left ( x+a \right ) } \left ( \left ( {\frac {a-x}{x+a}} \right ) ^{{\frac {1}{2\,a}\sqrt {{a}^{2}-{b}^{2}}}}{\it \_C1}+ \left ( {\frac {a-x}{x+a}} \right ) ^{-{\frac {1}{2\,a}\sqrt {{a}^{2}-{b}^{2}}}}{\it \_C2} \right ) \right \} \]