Internal problem ID [8944]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1365.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Halm]
Solve \begin {gather*} \boxed {y^{\prime \prime }+\frac {a y}{\left (x^{2}+1\right )^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 65
dsolve(diff(diff(y(x),x),x) = -a/(x^2+1)^2*y(x),y(x), singsol=all)
\[ y \relax (x ) = c_{1} \sqrt {x^{2}+1}\, \left (\frac {x +i}{-x +i}\right )^{\frac {\sqrt {a +1}}{2}}+c_{2} \sqrt {x^{2}+1}\, \left (\frac {x +i}{-x +i}\right )^{-\frac {\sqrt {a +1}}{2}} \]
✓ Solution by Mathematica
Time used: 0.046 (sec). Leaf size: 83
DSolve[y''[x] == -((a*y[x])/(1 + x^2)^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{2} \sqrt {x^2+1} e^{i \sqrt {a+1} \text {ArcTan}(x)} \left (\frac {i c_2 (1-i x)^{\sqrt {a+1}} (1+i x)^{-\sqrt {a+1}}}{\sqrt {a+1}}+2 c_1\right ) \\ \end{align*}