54.2.56 problem 632

Internal problem ID [11930]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 632
Date solved : Tuesday, September 30, 2025 at 11:44:38 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*}

✓ Maple. Time used: 0.069 (sec). Leaf size: 52

ode:=diff(y(x),x) = 1/(y(x)*exp(-x)+1)*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ x +\frac {\ln \left (y^{2} {\mathrm e}^{-2 x}+y \,{\mathrm e}^{-x}-1\right )}{2}-\frac {\sqrt {5}\, \operatorname {arctanh}\left (\frac {2 y \sqrt {5}\, {\mathrm e}^{-x}}{5}+\frac {\sqrt {5}}{5}\right )}{5}-c_1 = 0 \]

✓ Mathematica. Time used: 0.098 (sec). Leaf size: 65

ode=D[y[x],x] == E^x/(1 + y[x]/E^x); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 3.510 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - exp(x)/(y(x)*exp(-x) + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} + x + \frac {\left (5 - \sqrt {5}\right ) \log {\left (y{\left (x \right )} e^{- x} + \frac {1}{2} + \frac {\sqrt {5}}{2} \right )}}{10} + \frac {\left (\sqrt {5} + 5\right ) \log {\left (y{\left (x \right )} e^{- x} - \frac {\sqrt {5}}{2} + \frac {1}{2} \right )}}{10} = 0 \]