87.29.18 problem 18

Internal problem ID [23898]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 8. Nonlinear differential equations and systems. Exercise at page 304
Problem number : 18
Date solved : Sunday, October 12, 2025 at 05:55:16 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )-x_{1} \left (t \right )^{3} \end{align*}

✓ Maple. Time used: 0.049 (sec). Leaf size: 51

ode:=[diff(x__1(t),t) = x__2(t), diff(x__2(t),t) = x__1(t)-x__1(t)^3]; 
dsolve(ode);
 

\begin{align*} \left \{x_{1} \left (t \right ) &= c_2 \sqrt {2}\, \sqrt {\frac {1}{c_2^{2}+1}}\, \operatorname {JacobiSN}\left (\left (\frac {i \sqrt {2}\, t}{2}+c_1 \right ) \sqrt {2}\, \sqrt {\frac {1}{c_2^{2}+1}}, c_2\right )\right \} \\ \{x_{2} \left (t \right ) &= \frac {d}{d t}x_{1} \left (t \right )\} \\ \end{align*}

✓ Mathematica. Time used: 0.173 (sec). Leaf size: 1092

ode={D[x1[t],t]==x2[t],D[x2[t],t]==x1[t]-x1[t]^3}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-x2(t) + Derivative(x1(t), t),0),Eq(x1(t)**3 - x1(t) + Derivative(x2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

NotImplementedError :