80.8.21 problem 27

Internal problem ID [21378]
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 : 27
Date solved : Thursday, October 02, 2025 at 07:30:13 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} x^{\prime \prime }+x+\frac {x^{2}}{3}&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.287 (sec). Leaf size: 57

ode:=diff(diff(x(t),t),t)+x(t)+1/3*x(t)^2 = 0; 
ic:=[x(0) = 1, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\begin{align*} x &= \operatorname {RootOf}\left (3 \int _{1}^{\textit {\_Z}}\frac {1}{\sqrt {-2 \textit {\_a}^{3}-9 \textit {\_a}^{2}+11}}d \textit {\_a} +t \right ) \\ x &= \operatorname {RootOf}\left (3 \int _{\textit {\_Z}}^{1}\frac {1}{\sqrt {-2 \textit {\_a}^{3}-9 \textit {\_a}^{2}+11}}d \textit {\_a} +t \right ) \\ \end{align*}

✗ Mathematica

ode=D[x[t],{t,2}]+x[t]+1/3*x[t]^2==0; 
ic={x[0]==1,Derivative[1][x][0] ==0}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

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✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t)**2/3 + x(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 1, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

Timed Out