ODE No. 1377

$$y^{″}(x)=-\frac{b^{2}y(x)}{{(a^2+x^2)}^2}$$ ✓ 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]
 

$$\left\{\{y(x)\to \frac{i{c}_2\sqrt{a^2+x^2}{(1-\frac{ix}{a})}^{\sqrt{\frac{a^2+b^2}{a^2}}}{(1+\frac{ix}{a})}^{-\sqrt{\frac{a^2+b^2}{a^2}}}{e}^{i\sqrt{\frac{a^2+b^2}{a^2}}{\tan }^{-1}\left(\frac{x}{a}\right)}}{2a\sqrt{\frac{a^2+b^2}{a^2}}}+c_1\sqrt{a^2+x^2}{e}^{i\sqrt{\frac{b^2}{a^2}+1}{\tan }^{-1}\left(\frac{x}{a}\right)}\}\right\}$$ ✓ 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))
 

$$y\left(x\right)=\sqrt{a^2+x^2}({\left(\frac{ix-a}{ix+a}\right)}^{\frac{\sqrt{a^2+b^2}}{2a}}{c}_1+{\left(\frac{ix-a}{ix+a}\right)}^{-\frac{\sqrt{a^2+b^2}}{2a}}{c}_2)$$