Internal problem ID [8990]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1411.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {y^{\prime \prime }-\frac {y}{{\mathrm e}^{x}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 27
dsolve(diff(diff(y(x),x),x) = 1/(exp(x)+1)*y(x),y(x), singsol=all)
\[ y \relax (x ) = \left (\left (\frac {1}{{\mathrm e}^{x}+1}+\ln \left ({\mathrm e}^{x}+1\right )\right ) c_{1}+c_{2}\right ) \left (1+{\mathrm e}^{-x}\right ) \]
✓ Solution by Mathematica
Time used: 0.205 (sec). Leaf size: 36
DSolve[y''[x] == y[x]/(1 + E^x),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to e^{-x} \left (c_1 \left (e^x+1\right )+c_2 \left (e^x+1\right ) \log \left (e^x+1\right )+c_2\right ) \\ \end{align*}