64.25.7 problem 7

Internal problem ID [13631]
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 : 7
Date solved : Monday, March 31, 2025 at 08:03:19 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} x^{\prime \prime }+\left (x^{2}+1\right ) x^{\prime }+x^{3}&=0 \end{align*}

✗ Maple

ode:=diff(diff(x(t),t),t)+(x(t)^2+1)*diff(x(t),t)+x(t)^3 = 0; 
dsolve(ode,x(t), singsol=all);
 

\[ \text {No solution found} \]

✗ Mathematica

ode=D[x[t],{t,2}]+(x[t]^2+1)*D[x[t],t]+x[t]^3==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)**2 + 1)*Derivative(x(t), t) + x(t)**3 + Derivative(x(t), (t, 2)),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

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