60.2.91 problem 667
Internal
problem
ID
[10665]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
1,
Additional
non-linear
first
order
Problem
number
:
667
Date
solved
:
Wednesday, March 05, 2025 at 12:17:01 PM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class C`]]
\begin{align*} y^{\prime }&=\frac {y^{3} {\mathrm e}^{-2 b x}}{y \,{\mathrm e}^{-b x}+1} \end{align*}
✓ Maple. Time used: 0.382 (sec). Leaf size: 94
ode:=diff(y(x),x) = y(x)^3/(y(x)*exp(-b*x)+1)*exp(-2*b*x);
dsolve(ode,y(x), singsol=all);
\[
-\frac {-\ln \left (y \,{\mathrm e}^{-b x}\right ) \sqrt {b \left (4+b \right )}+\left (\frac {\ln \left (-b y \,{\mathrm e}^{-b x}+y^{2} {\mathrm e}^{-2 b x}-b \right )}{2}-b x +c_{1} \right ) \sqrt {b \left (4+b \right )}+b \,\operatorname {arctanh}\left (\frac {-2 y \,{\mathrm e}^{-b x}+b}{\sqrt {b \left (4+b \right )}}\right )}{\sqrt {b \left (4+b \right )}} = 0
\]
✓ Mathematica. Time used: 0.526 (sec). Leaf size: 130
ode=D[y[x],x] == y[x]^3/(E^(2*b*x)*(1 + y[x]/E^(b*x)));
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\int _1^{-\frac {e^{2 b x} \left (e^{b x} b+(b+3) y(x)\right )}{\sqrt [3]{-b^2 (2 b+9) e^{6 b x}} \left (y(x)+e^{b x}\right )}}\frac {1}{K[1]^3+\frac {3 \sqrt [3]{-1} (b+3) K[1]}{\sqrt [3]{b} (2 b+9)^{2/3}}+1}dK[1]=\frac {1}{9} x e^{-4 b x} \left (-b^2 (2 b+9) e^{6 b x}\right )^{2/3}+c_1,y(x)\right ]
\]
✓ Sympy. Time used: 5.513 (sec). Leaf size: 304
from sympy import *
x = symbols("x")
b = symbols("b")
y = Function("y")
ode = Eq(Derivative(y(x), x) - y(x)**3*exp(-2*b*x)/(y(x)*exp(-b*x) + 1),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
C_{1} + b x - \frac {\left (- \frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right ) \log {\left (y{\left (x \right )} e^{- b x} + \frac {- \frac {b^{2} \left (- \frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )^{2}}{4} - \frac {b^{2} \left (- \frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )}{2} - \frac {5 b \left (- \frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )^{2}}{2} - \frac {7 b \left (- \frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )}{2} + 3 b + \frac {6 \sqrt {b \left (b + 4\right )}}{b + 4} - 6 \left (- \frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )^{2} + 6}{2 b + 9} \right )}}{2} - \frac {\left (\frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right ) \log {\left (y{\left (x \right )} e^{- b x} + \frac {- \frac {b^{2} \left (\frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )^{2}}{4} - \frac {b^{2} \left (\frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )}{2} - \frac {5 b \left (\frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )^{2}}{2} - \frac {7 b \left (\frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )}{2} + 3 b - \frac {6 \sqrt {b \left (b + 4\right )}}{b + 4} - 6 \left (\frac {\sqrt {b \left (b + 4\right )}}{b + 4} + 1\right )^{2} + 6}{2 b + 9} \right )}}{2} + \log {\left (y{\left (x \right )} e^{- b x} \right )} = 0
\]