54.2.54 problem 630

Internal problem ID [11928]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 630
Date solved : Tuesday, September 30, 2025 at 11:44:31 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*}

✓ Maple. Time used: 0.095 (sec). Leaf size: 58

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

\[ y = \operatorname {RootOf}\left (-{\mathrm e}^{\operatorname {RootOf}\left (4 \,{\mathrm e}^{\textit {\_Z}} \cosh \left (\frac {\sqrt {b \left (4+b \right )}\, \left (2 c_1 b -2 b x -\textit {\_Z} \right )}{2 b}\right )^{2}+b +4\right )}-1+b \textit {\_Z} +b \,\textit {\_Z}^{2}\right ) {\mathrm e}^{b x} \]

✓ Mathematica. Time used: 0.191 (sec). Leaf size: 101

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

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

✓ Sympy. Time used: 9.796 (sec). Leaf size: 124

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

\[ C_{1} + b x + \frac {\left (- \frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right ) \log {\left (y{\left (x \right )} e^{- b x} + \frac {- \frac {b \left (- \frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )}{2} + b + \frac {2 \sqrt {b \left (b + 4\right )}}{b + 4}}{b} \right )}}{2} + \frac {\left (\frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right ) \log {\left (y{\left (x \right )} e^{- b x} + \frac {- \frac {b \left (\frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )}{2} + b - \frac {2 \sqrt {b \left (b + 4\right )}}{b + 4}}{b} \right )}}{2} = 0 \]