64.1.15 problem 2.4 (ii)

Internal problem ID [15592]
Book : Nonlinear Ordinary Differential Equations by D.W.Jordna and P.Smith. 4th edition 1999. Oxford Univ. Press. NY
Section : Chapter 2. Plane autonomous systems and linearization. Problems page 79
Problem number : 2.4 (ii)
Date solved : Thursday, October 02, 2025 at 10:20:32 AM
CAS classification : [[_2nd_order, _missing_x], _Duffing, [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} x^{\prime \prime }+x+x^{3}&=0 \end{align*}

✓ Maple. Time used: 0.015 (sec). Leaf size: 55

ode:=diff(diff(x(t),t),t)+x(t)+x(t)^3 = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ x = c_2 \,\operatorname {JacobiSN}\left (\frac {\left (\sqrt {2}\, \sqrt {3}\, t +2 c_1 \right ) \sqrt {2}\, \sqrt {-\frac {1}{c_2^{2}-3}}}{2}, \frac {i c_2 \sqrt {3}}{3}\right ) \sqrt {2}\, \sqrt {-\frac {1}{c_2^{2}-3}} \]

✓ Mathematica. Time used: 60.086 (sec). Leaf size: 169

ode=D[x[t],{t,2}]+x[t]+x[t]^3==0; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

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