\[ y''(x)=-\frac {b^2 y(x)}{\left (a^2+x^2\right )^2} \] ✓ Mathematica : cpu = 0.187499 (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 = 0.128 (sec), leaf count = 83
\[\left \{y \left (x \right ) = \sqrt {a^{2}+x^{2}}\, \left (c_{1} \left (\frac {i x -a}{i x +a}\right )^{\frac {\sqrt {a^{2}+b^{2}}}{2 a}}+c_{2} \left (\frac {i x -a}{i x +a}\right )^{-\frac {\sqrt {a^{2}+b^{2}}}{2 a}}\right )\right \}\]