60.2.54 problem 630
Internal
problem
ID
[10628]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
630
Date
solved
:
Wednesday, March 05, 2025 at 12:11:45 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: 0.336 (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);
\[
y = \operatorname {RootOf}\left (-{\mathrm e}^{\operatorname {RootOf}\left (4 \,{\mathrm e}^{\textit {\_Z}} \cosh \left (\frac {\sqrt {b \left (4+b \right )}\, \left (2 c_{1} b -2 b x -\textit {\_Z} \right )}{2 b}\right )^{2}+b +4\right )}-1+\textit {\_Z} b +b \,\textit {\_Z}^{2}\right ) {\mathrm e}^{b x}
\]
✓ Mathematica. Time used: 0.322 (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: 13.788 (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)
\[
C_{1} + b x + \frac {\left (- \frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right ) \log {\left (y{\left (x \right )} e^{- b x} + \frac {- \frac {b \left (- \frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )}{2} + b + \frac {2 \sqrt {b \left (b + 4\right )}}{b + 4}}{b} \right )}}{2} + \frac {\left (\frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right ) \log {\left (y{\left (x \right )} e^{- b x} + \frac {- \frac {b \left (\frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )}{2} + b - \frac {2 \sqrt {b \left (b + 4\right )}}{b + 4}}{b} \right )}}{2} = 0
\]