80.11.21 problem 21
Internal
problem
ID
[21429]
Book
:
A
Textbook
on
Ordinary
Differential
Equations
by
Shair
Ahmad
and
Antonio
Ambrosetti.
Second
edition.
ISBN
978-3-319-16407-6.
Springer
2015
Section
:
Chapter
12.
Stability
theory.
Excercise
12.6
at
page
270
Problem
number
:
21
Date
solved
:
Thursday, October 02, 2025 at 07:31:28 PM
CAS
classification
:
[_quadrature]
\begin{align*} x^{\prime }&=\lambda x-x^{5} \end{align*}
✓ Maple. Time used: 0.010 (sec). Leaf size: 177
ode:=diff(x(t),t) = lambda*x(t)-x(t)^5;
dsolve(ode,x(t), singsol=all);
\begin{align*}
x &= \frac {\sqrt {\left (c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1\right ) \sqrt {\left (c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1\right ) \lambda }}}{c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1} \\
x &= \frac {\sqrt {-\left (c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1\right ) \sqrt {\left (c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1\right ) \lambda }}}{c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1} \\
x &= -\frac {\sqrt {\left (c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1\right ) \sqrt {\left (c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1\right ) \lambda }}}{c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1} \\
x &= -\frac {\sqrt {-\left (c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1\right ) \sqrt {\left (c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1\right ) \lambda }}}{c_1 \lambda \,{\mathrm e}^{-4 \lambda t}+1} \\
\end{align*}
✓ Mathematica. Time used: 1.507 (sec). Leaf size: 207
ode=D[x[t],t]==\[Lambda]*x[t]-x[t]^5;
ic={};
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
\begin{align*} x(t)&\to -\frac {\sqrt [4]{\lambda } e^{\lambda (t-c_1)}}{\sqrt [4]{-1+e^{4 \lambda (t-c_1)}}}\\ x(t)&\to -\frac {i \sqrt [4]{\lambda } e^{\lambda (t-c_1)}}{\sqrt [4]{-1+e^{4 \lambda (t-c_1)}}}\\ x(t)&\to \frac {i \sqrt [4]{\lambda } e^{\lambda (t-c_1)}}{\sqrt [4]{-1+e^{4 \lambda (t-c_1)}}}\\ x(t)&\to \frac {\sqrt [4]{\lambda } e^{\lambda (t-c_1)}}{\sqrt [4]{-1+e^{4 \lambda (t-c_1)}}}\\ x(t)&\to 0\\ x(t)&\to -\sqrt [4]{\lambda }\\ x(t)&\to -i \sqrt [4]{\lambda }\\ x(t)&\to i \sqrt [4]{\lambda }\\ x(t)&\to \sqrt [4]{\lambda } \end{align*}
✓ Sympy. Time used: 4.531 (sec). Leaf size: 116
from sympy import *
t = symbols("t")
x = Function("x")
ode = Eq(-lambda_*x(t) + x(t)**5 + Derivative(x(t), t),0)
ics = {}
dsolve(ode,func=x(t),ics=ics)
\[
\left [ x{\left (t \right )} = - i \sqrt [4]{- \frac {\lambda _{} e^{4 \lambda _{} t}}{e^{C_{1} \lambda _{}} - e^{4 \lambda _{} t}}}, \ x{\left (t \right )} = i \sqrt [4]{- \frac {\lambda _{} e^{4 \lambda _{} t}}{e^{C_{1} \lambda _{}} - e^{4 \lambda _{} t}}}, \ x{\left (t \right )} = - \sqrt [4]{- \frac {\lambda _{} e^{4 \lambda _{} t}}{e^{C_{1} \lambda _{}} - e^{4 \lambda _{} t}}}, \ x{\left (t \right )} = \sqrt [4]{- \frac {\lambda _{} e^{4 \lambda _{} t}}{e^{C_{1} \lambda _{}} - e^{4 \lambda _{} t}}}\right ]
\]