\[ y''(x)=-\frac {b^2 y(x)}{\left (a^2+x^2\right )^2} \] ✓ Mathematica : cpu = 0.258666 (sec), leaf count = 97
\[\left \{\left \{y(x)\to \frac {1}{2} \sqrt {a^2+x^2} e^{-i \sqrt {\frac {b^2}{a^2}+1} \tan ^{-1}\left (\frac {x}{a}\right )} \left (2 c_1 e^{2 i \sqrt {\frac {b^2}{a^2}+1} \tan ^{-1}\left (\frac {x}{a}\right )}+\frac {i c_2}{a \sqrt {\frac {b^2}{a^2}+1}}\right )\right \}\right \}\]
✓ Maple : cpu = 0.129 (sec), leaf count = 83
\[ \left \{ y \left ( x \right ) =\sqrt {{a}^{2}+{x}^{2}} \left ( \left ( {\frac {ix-a}{ix+a}} \right ) ^{-{\frac {1}{2\,a}\sqrt {{a}^{2}+{b}^{2}}}}{\it \_C2}+ \left ( {\frac {ix-a}{ix+a}} \right ) ^{{\frac {1}{2\,a}\sqrt {{a}^{2}+{b}^{2}}}}{\it \_C1} \right ) \right \} \]