80.8.13 problem 17

Internal problem ID [21370]
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 8. Qualitative analysis of 2 by 2 systems and nonlinear second order equations. Excercise 8.5 at page 184
Problem number : 17
Date solved : Thursday, October 02, 2025 at 07:29:49 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 45

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

\[ x = c_2 \sqrt {2}\, \sqrt {\frac {1}{c_2^{2}+1}}\, \operatorname {JacobiSN}\left (\frac {\left (i \sqrt {2}\, t +2 c_1 \right ) \sqrt {2}\, \sqrt {\frac {1}{c_2^{2}+1}}}{2}, c_2\right ) \]

✓ Mathematica. Time used: 60.1 (sec). Leaf size: 173

ode=D[x[t],{t,2}]==x[t]-x[t]^3; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to -\frac {i \text {sn}\left (\frac {\sqrt {-\left (\left (\sqrt {2 c_1+1}+1\right ) (t+c_2){}^2\right )}}{\sqrt {2}}|\frac {1-\sqrt {2 c_1+1}}{\sqrt {2 c_1+1}+1}\right )}{\sqrt {\frac {1}{-1+\sqrt {1+2 c_1}}}}\\ x(t)&\to \frac {i \text {sn}\left (\frac {\sqrt {-\left (\left (\sqrt {2 c_1+1}+1\right ) (t+c_2){}^2\right )}}{\sqrt {2}}|\frac {1-\sqrt {2 c_1+1}}{\sqrt {2 c_1+1}+1}\right )}{\sqrt {\frac {1}{-1+\sqrt {1+2 c_1}}}} \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t)**3 - x(t) + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

Timed Out