Internal problem ID [8956]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1377.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_Emden, _Fowler]]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {b^{2} y}{\left (a^{2}+x^{2}\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.008 (sec). Leaf size: 91
dsolve(diff(diff(y(x),x),x) = -b^2/(a^2+x^2)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {a^{2}+x^{2}}\, \left (\frac {i x -a}{i x +a}\right )^{\frac {\sqrt {a^{2}+b^{2}}}{2 a}}+c_{2} \sqrt {a^{2}+x^{2}}\, \left (\frac {i x -a}{i x +a}\right )^{-\frac {\sqrt {a^{2}+b^{2}}}{2 a}} \]
✓ Solution by Mathematica
Time used: 0.122 (sec). Leaf size: 123
DSolve[y''[x] == -((b^2*y[x])/(a^2 + x^2)^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {a^2+x^2} e^{i \sqrt {\frac {b^2}{a^2}+1} \text {ArcTan}\left (\frac {x}{a}\right )} \left (\frac {i c_2 \left (1-\frac {i x}{a}\right )^{\sqrt {\frac {b^2}{a^2}+1}} \left (1+\frac {i x}{a}\right )^{-\sqrt {\frac {b^2}{a^2}+1}}}{\sqrt {\frac {b^2}{a^2}+1}}+2 a c_1\right )}{2 a} \\ \end{align*}