60.2.92 problem 668

Internal problem ID [10679]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 668
Date solved : Monday, January 27, 2025 at 09:25:08 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*}

✓ Solution by Maple

Time used: 1.697 (sec). Leaf size: 57

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

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

✓ Solution by Mathematica

Time used: 0.468 (sec). Leaf size: 99

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

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