64.25.4 problem 4

Internal problem ID [13628]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 13, Limit cycles and periodic solutions. Section 13.4, Exercises page 706
Problem number : 4
Date solved : Monday, March 31, 2025 at 08:03:08 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✗ Maple

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

\[ \text {No solution found} \]

✗ Mathematica

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

Not solved

✗ Sympy

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

NotImplementedError : The given ODE (sqrt((27*x(t) + 27*Derivative(x(t), (t, 2)))**2 + 108)/2 + 27*x(t)/2 + 27*Derivative(x(t), (t, 2))/2)**(1/3)/3 + Derivative(x(t), t) - 1/(sqrt((27*x(t) + 27*Derivative(x(t), (t, 2)))**2 + 108)/2 + 27*x(t)/2 + 27*Derivative(x(t), (t, 2))/2)**(1/3) cannot be solved by the factorable group method