Internal problem ID [8212]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 632.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], [_Abel, 2nd type, class C]]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {{\mathrm e}^{x}}{y \,{\mathrm e}^{-x}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.648 (sec). Leaf size: 52
dsolve(diff(y(x),x) = 1/(y(x)*exp(-x)+1)*exp(x),y(x), singsol=all)
\[ x -\frac {\sqrt {5}\, \arctanh \left (\frac {\left (2 y \relax (x ) {\mathrm e}^{-x}+1\right ) \sqrt {5}}{5}\right )}{5}+\frac {\ln \left (y \relax (x )^{2} {\mathrm e}^{-2 x}+y \relax (x ) {\mathrm e}^{-x}-1\right )}{2}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.161 (sec). Leaf size: 65
DSolve[y'[x] == E^x/(1 + y[x]/E^x),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} \log \left (-e^{-2 x} y(x)^2-e^{-x} y(x)+1\right )+x=\frac {\tanh ^{-1}\left (\frac {y(x)+3 e^x}{\sqrt {5} \left (y(x)+e^x\right )}\right )}{\sqrt {5}}+c_1,y(x)\right ] \]