Internal problem ID [9738]
Internal file name [OUTPUT/8680_Monday_June_06_2022_05_10_39_AM_34401268/index.tex]
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.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }-\frac {y}{{\mathrm e}^{x}+1}=0} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] <- linear_1 successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 27
dsolve(diff(diff(y(x),x),x) = 1/(exp(x)+1)*y(x),y(x), singsol=all)
\[ y \left (x \right ) = {\mathrm e}^{-x} \left (c_{1} \left ({\mathrm e}^{x}+1\right ) \ln \left ({\mathrm e}^{x}+1\right )+{\mathrm e}^{x} c_{2} +c_{1} +c_{2} \right ) \]
✓ Solution by Mathematica
Time used: 0.456 (sec). Leaf size: 36
DSolve[y''[x] == y[x]/(1 + E^x),y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to e^{-x} \left (c_1 \left (e^x+1\right )+c_2 \left (e^x+1\right ) \log \left (e^x+1\right )+c_2\right ) \]