54.2.92 problem 668

Internal problem ID [11966]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 668
Date solved : Tuesday, September 30, 2025 at 11:49:43 PM
CAS classification : [[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class C`]]

\begin{align*} y^{\prime }&=\frac {y^{3} {\mathrm e}^{-2 x}}{y \,{\mathrm e}^{-x}+1} \end{align*}

✓ Maple. Time used: 0.481 (sec). Leaf size: 57

ode:=diff(y(x),x) = 1/(y(x)*exp(-x)+1)*y(x)^3*exp(-2*x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (2 \sqrt {5}\, \operatorname {arctanh}\left (\frac {\sqrt {5}\, \left (-1+2 \,{\mathrm e}^{\textit {\_Z} -x}\right )}{5}\right )-5 \ln \left ({\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{\textit {\_Z} +x}-{\mathrm e}^{2 x}\right )-10 c_1 +10 \textit {\_Z} +10 x \right )} \]

✓ Mathematica. Time used: 0.291 (sec). Leaf size: 99

ode=D[y[x],x] == y[x]^3/(E^(2*x)*(1 + y[x]/E^x)); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {e^{8 x} \left (4 y(x)+e^x\right )}{\sqrt [3]{11} \left (-e^{6 x}\right )^{4/3} \left (y(x)+e^x\right )}}\frac {1}{K[1]^3+\frac {12 \sqrt [3]{-1} K[1]}{11^{2/3}}+1}dK[1]=\frac {1}{9} 11^{2/3} e^{-4 x} \left (-e^{6 x}\right )^{2/3} x+c_1,y(x)\right ] \]

✓ Sympy. Time used: 5.026 (sec). Leaf size: 94

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - y(x)**3*exp(-2*x)/(y(x)*exp(-x) + 1),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ C_{1} + x + \log {\left (y{\left (x \right )} e^{- x} \right )} - \frac {\left (5 - \sqrt {5}\right ) \log {\left (y{\left (x \right )} e^{- x} - \frac {7 \left (5 - \sqrt {5}\right )^{2}}{220} + \frac {2 \sqrt {5}}{11} + \frac {5}{11} \right )}}{10} - \frac {\left (\sqrt {5} + 5\right ) \log {\left (y{\left (x \right )} e^{- x} - \frac {7 \left (\sqrt {5} + 5\right )^{2}}{220} - \frac {2 \sqrt {5}}{11} + \frac {5}{11} \right )}}{10} = 0 \]