\[ y''(x)=-\frac {b^2 y(x)}{\left (a^2+x^2\right )^2} \] ✓ Mathematica : cpu = 0.193292 (sec), leaf count = 163
\[\left \{\left \{y(x)\to \frac {i c_2 \sqrt {a^2+x^2} \left (1-\frac {i x}{a}\right )^{\sqrt {\frac {a^2+b^2}{a^2}}} \left (1+\frac {i x}{a}\right )^{-\sqrt {\frac {a^2+b^2}{a^2}}} e^{i \sqrt {\frac {a^2+b^2}{a^2}} \tan ^{-1}\left (\frac {x}{a}\right )}}{2 a \sqrt {\frac {a^2+b^2}{a^2}}}+c_1 \sqrt {a^2+x^2} e^{i \sqrt {\frac {b^2}{a^2}+1} \tan ^{-1}\left (\frac {x}{a}\right )}\right \}\right \}\] ✓ Maple : cpu = 0.252 (sec), leaf count = 83
\[ \left \{ y \left ( x \right ) =\sqrt {{a}^{2}+{x}^{2}} \left ( \left ( {\frac {ix-a}{ix+a}} \right ) ^{-{\frac {1}{2\,a}\sqrt {{a}^{2}+{b}^{2}}}}{\it \_C2}+ \left ( {\frac {ix-a}{ix+a}} \right ) ^{{\frac {1}{2\,a}\sqrt {{a}^{2}+{b}^{2}}}}{\it \_C1} \right ) \right \} \]