60.2.54 problem 630

Internal problem ID [10641]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 630
Date solved : Monday, January 27, 2025 at 09:20:04 PM
CAS classification : [[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class C`]]

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

✓ Solution by Maple

Time used: 0.336 (sec). Leaf size: 58

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

\[ y = \operatorname {RootOf}\left (-{\mathrm e}^{\operatorname {RootOf}\left (4 \,{\mathrm e}^{\textit {\_Z}} \cosh \left (\frac {\sqrt {b \left (4+b \right )}\, \left (2 c_{1} b -2 b x -\textit {\_Z} \right )}{2 b}\right )^{2}+b +4\right )}-1+\textit {\_Z} b +b \,\textit {\_Z}^{2}\right ) {\mathrm e}^{b x} \]

✓ Solution by Mathematica

Time used: 0.322 (sec). Leaf size: 101

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

\[ \text {Solve}\left [\frac {1}{2} b \left (\log \left (-b e^{-2 b x} y(x)^2-b e^{-b x} y(x)+1\right )+2 b x\right )=\frac {b \arctan \left (\frac {(b+2) \left (-e^{b x}\right )-b y(x)}{b \sqrt {-\frac {b+4}{b}} \left (e^{b x}+y(x)\right )}\right )}{\sqrt {-\frac {b+4}{b}}}+c_1,y(x)\right ] \]