60.2.56 problem 632

Internal problem ID [10643]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 632
Date solved : Monday, January 27, 2025 at 09:20:12 PM
CAS classification : [[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {{\mathrm e}^{x}}{y \,{\mathrm e}^{-x}+1} \end{align*}

Solution by Maple

Time used: 0.347 (sec). Leaf size: 52

dsolve(diff(y(x),x) = 1/(y(x)*exp(-x)+1)*exp(x),y(x), singsol=all)
 
\[ x -\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {2 y \sqrt {5}\, {\mathrm e}^{-x}}{5}+\frac {\sqrt {5}}{5}\right )}{5}+\frac {\ln \left (y^{2} {\mathrm e}^{-2 x}+y \,{\mathrm e}^{-x}-1\right )}{2}-c_{1} = 0 \]

Solution by Mathematica

Time used: 0.159 (sec). Leaf size: 65

DSolve[D[y[x],x] == E^x/(1 + y[x]/E^x),y[x],x,IncludeSingularSolutions -> True]
 
\[ \text {Solve}\left [\frac {1}{2} \log \left (-e^{-2 x} y(x)^2-e^{-x} y(x)+1\right )+x=\frac {\text {arctanh}\left (\frac {y(x)+3 e^x}{\sqrt {5} \left (y(x)+e^x\right )}\right )}{\sqrt {5}}+c_1,y(x)\right ] \]