55.26.23 problem 23
Internal
problem
ID
[13770]
Book
:
Handbook
of
exact
solutions
for
ordinary
differential
equations.
By
Polyanin
and
Zaitsev.
Second
edition
Section
:
Chapter
1,
section
1.4.
Equations
Containing
Polynomial
Functions
of
y.
subsection
1.4.1-2
Abel
equations
of
the
first
kind.
Problem
number
:
23
Date
solved
:
Thursday, October 02, 2025 at 08:06:23 AM
CAS
classification
:
[[_1st_order, _with_linear_symmetries], _Abel]
\begin{align*} y^{\prime }&=-\frac {{\mathrm e}^{2 \lambda x} y^{3}}{3 \lambda }+\frac {2 \lambda ^{2} {\mathrm e}^{-\lambda x}}{3} \end{align*}
✓ Maple. Time used: 0.038 (sec). Leaf size: 64
ode:=diff(y(x),x) = -1/3/lambda*exp(2*lambda*x)*y(x)^3+2/3*lambda^2*exp(-lambda*x);
dsolve(ode,y(x), singsol=all);
\[
y = -\frac {2 \,{\mathrm e}^{-\lambda x} \lambda \left ({\mathrm e}^{\operatorname {RootOf}\left (6 \lambda x \,{\mathrm e}^{\textit {\_Z}}-2 \ln \left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right ) {\mathrm e}^{\textit {\_Z}}+6 c_1 \,{\mathrm e}^{\textit {\_Z}}+2 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}-27 \lambda x +9 \ln \left (2 \,{\mathrm e}^{\textit {\_Z}}-9\right )-27 c_1 -9 \textit {\_Z} -9\right )}-3\right )}{3}
\]
✓ Mathematica. Time used: 2.031 (sec). Leaf size: 254
ode=D[y[x],x]==-1/(3*\[Lambda])*Exp[2*\[Lambda]*x]*y[x]^3+2/3*\[Lambda]^2*Exp[-\[Lambda]*x];
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
\text {Solve}\left [\frac {\sqrt [3]{2} \left (\frac {\sqrt [3]{-\frac {e^{3 \lambda x}}{\lambda ^3}} y(x)}{\sqrt [3]{2}}+2^{2/3}\right ) \left (2^{2/3}-2^{2/3} \sqrt [3]{-\frac {e^{3 \lambda x}}{\lambda ^3}} y(x)\right ) \left (\left (1-\sqrt [3]{-\frac {e^{3 \lambda x}}{\lambda ^3}} y(x)\right ) \log \left (2^{2/3}-2^{2/3} \sqrt [3]{-\frac {e^{3 \lambda x}}{\lambda ^3}} y(x)\right )+\left (\sqrt [3]{-\frac {e^{3 \lambda x}}{\lambda ^3}} y(x)-1\right ) \log \left (2 \left (\frac {\sqrt [3]{-\frac {e^{3 \lambda x}}{\lambda ^3}} y(x)}{\sqrt [3]{2}}+2^{2/3}\right )\right )-3\right )}{9 \left (\frac {e^{3 \lambda x} y(x)^3}{\lambda ^3}+3 \sqrt [3]{-\frac {e^{3 \lambda x}}{\lambda ^3}} y(x)-2\right )}=\frac {1}{3} 2^{2/3} \lambda ^2 x e^{\lambda (-x)} \sqrt [3]{-\frac {e^{3 \lambda x}}{\lambda ^3}}+c_1,y(x)\right ]
\]
✓ Sympy. Time used: 2.638 (sec). Leaf size: 48
from sympy import *
x = symbols("x")
lambda_ = symbols("lambda_")
y = Function("y")
ode = Eq(-2*lambda_**2*exp(-lambda_*x)/3 + Derivative(y(x), x) + y(x)**3*exp(2*lambda_*x)/(3*lambda_),0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
C_{1} - \lambda _{} x - \frac {\lambda _{} \left (\frac {3}{\lambda _{} + y{\left (x \right )} e^{\lambda _{} x}} + \frac {\log {\left (- 2 \lambda _{} + y{\left (x \right )} e^{\lambda _{} x} \right )} - \log {\left (\lambda _{} + y{\left (x \right )} e^{\lambda _{} x} \right )}}{\lambda _{}}\right )}{3} = 0
\]