Internal problem ID [8210]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 630.
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 {{\mathrm e}^{b x}}{y \,{\mathrm e}^{-b x}+1}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.417 (sec). Leaf size: 98
dsolve(diff(y(x),x) = 1/(y(x)*exp(-b*x)+1)*exp(b*x),y(x), singsol=all)
\[ y \relax (x ) = \RootOf \left (-{\mathrm e}^{\RootOf \left (\left (\tanh ^{2}\left (\frac {\sqrt {b^{2}+4 b}\, \left (2 c_{1} b -2 b x -\textit {\_Z} \right )}{2 b}\right )\right ) b +4 \left (\tanh ^{2}\left (\frac {\sqrt {b^{2}+4 b}\, \left (2 c_{1} b -2 b x -\textit {\_Z} \right )}{2 b}\right )\right )-4 \,{\mathrm e}^{\textit {\_Z}}-b -4\right )}-1+\textit {\_Z} b +b \,\textit {\_Z}^{2}\right ) {\mathrm e}^{b x} \]
✓ Solution by Mathematica
Time used: 0.315 (sec). Leaf size: 101
DSolve[y'[x] == E^(b*x)/(1 + y[x]/E^(b*x)),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\frac {1}{2} b \left (\log \left (-b e^{-2 b x} y(x)^2-b e^{-b x} y(x)+1\right )+2 b x\right )=\frac {b \text {ArcTan}\left (\frac {(b+2) \left (-e^{b x}\right )-b y(x)}{b \sqrt {-\frac {b+4}{b}} \left (e^{b x}+y(x)\right )}\right )}{\sqrt {-\frac {b+4}{b}}}+c_1,y(x)\right ] \]