Internal problem ID [8248]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 668.
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 {y^{3} {\mathrm e}^{-2 x}}{y \,{\mathrm e}^{-x}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 34.125 (sec). Leaf size: 58
dsolve(diff(y(x),x) = 1/(y(x)*exp(-x)+1)*y(x)^3*exp(-2*x),y(x), singsol=all)
\[ y \relax (x ) = {\mathrm e}^{\RootOf \left (2 \sqrt {5}\, \arctanh \left (\frac {\left (-2 \,{\mathrm e}^{\textit {\_Z}}+{\mathrm e}^{x}\right ) \sqrt {5}\, {\mathrm e}^{-x}}{5}\right )+5 \ln \left ({\mathrm e}^{2 \textit {\_Z}}-{\mathrm e}^{\textit {\_Z} +x}-{\mathrm e}^{2 x}\right )+10 c_{1}-10 \textit {\_Z} -10 x \right )} \]
✓ Solution by Mathematica
Time used: 0.488 (sec). Leaf size: 73
DSolve[y'[x] == y[x]^3/(E^(2*x)*(1 + y[x]/E^x)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\log (y(x))+\frac {1}{10} \left (-\left (5+\sqrt {5}\right ) \log \left (-\sqrt {5} y(x)+y(x)+2 e^x\right )+\left (\sqrt {5}-5\right ) \log \left (\sqrt {5} y(x)+y(x)+2 e^x\right )+10 \log \left (e^x\right )\right )=c_1,y(x)\right ] \]