80.11.22 problem 22

Internal problem ID [21430]
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 : 22
Date solved : Thursday, October 02, 2025 at 07:31:29 PM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=\lambda x-x^{3}-x^{5} \end{align*}

✓ Maple. Time used: 0.052 (sec). Leaf size: 59

ode:=diff(x(t),t) = lambda*x(t)-x(t)^3-x(t)^5; 
dsolve(ode,x(t), singsol=all);
 

\[ t +\frac {-\frac {2 \,\operatorname {arctanh}\left (\frac {2 x^{2}+1}{\sqrt {4 \lambda +1}}\right )}{\sqrt {4 \lambda +1}}+\ln \left (x^{4}+x^{2}-\lambda \right )}{4 \lambda }-\frac {\ln \left (x\right )}{\lambda }+c_1 = 0 \]

✓ Mathematica. Time used: 0.292 (sec). Leaf size: 217

ode=D[x[t],t]==\[Lambda]*x[t]-x[t]^3-x[t]^5; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \text {InverseFunction}\left [\frac {\left (\sqrt {4 \lambda +1}+1\right ) \log \left (2 \text {$\#$1}^2-\sqrt {4 \lambda +1}+1\right )+\left (\sqrt {4 \lambda +1}-1\right ) \log \left (2 \text {$\#$1}^2+\sqrt {4 \lambda +1}+1\right )-4 \sqrt {4 \lambda +1} \log (\text {$\#$1})}{4 \lambda \sqrt {4 \lambda +1}}\&\right ][-t+c_1]\\ x(t)&\to 0\\ x(t)&\to -\frac {\sqrt {-\sqrt {4 \lambda +1}-1}}{\sqrt {2}}\\ x(t)&\to \frac {\sqrt {-\sqrt {4 \lambda +1}-1}}{\sqrt {2}}\\ x(t)&\to -\frac {\sqrt {\sqrt {4 \lambda +1}-1}}{\sqrt {2}}\\ x(t)&\to \frac {\sqrt {\sqrt {4 \lambda +1}-1}}{\sqrt {2}} \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-lambda_*x(t) + x(t)**5 + x(t)**3 + Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

TypeError : cannot determine truth value of Relational: 2 < 4*sqrt(4*_lambda_ + 1)