\[ y''(x)=-\frac {b^2 y(x)}{\left (a^2+x^2\right )^2} \] ✓ Mathematica : cpu = 0.16716 (sec), leaf count = 163
DSolve[Derivative[2][y][x] == -((b^2*y[x])/(a^2 + x^2)^2),y[x],x]
\[\left \{\left \{y(x)\to \frac {i c_2 \sqrt {a^2+x^2} \left (1-\frac {i x}{a}\right )^{\sqrt {\frac {a^2+b^2}{a^2}}} \left (1+\frac {i x}{a}\right )^{-\sqrt {\frac {a^2+b^2}{a^2}}} e^{i \sqrt {\frac {a^2+b^2}{a^2}} \tan ^{-1}\left (\frac {x}{a}\right )}}{2 a \sqrt {\frac {a^2+b^2}{a^2}}}+c_1 \sqrt {a^2+x^2} e^{i \sqrt {\frac {b^2}{a^2}+1} \tan ^{-1}\left (\frac {x}{a}\right )}\right \}\right \}\] ✓ Maple : cpu = 0.119 (sec), leaf count = 83
dsolve(diff(diff(y(x),x),x) = -b^2/(a^2+x^2)^2*y(x),y(x))
\[y \left (x \right ) = \sqrt {a^{2}+x^{2}}\, \left (\left (\frac {i x -a}{i x +a}\right )^{\frac {\sqrt {a^{2}+b^{2}}}{2 a}} c_{1}+\left (\frac {i x -a}{i x +a}\right )^{-\frac {\sqrt {a^{2}+b^{2}}}{2 a}} c_{2}\right )\]