80.8.23 problem 29

Internal problem ID [21380]
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 : 29
Date solved : Friday, October 03, 2025 at 07:51:21 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

With initial conditions

\begin{align*} x \left (0\right )&={\frac {1}{4}} \\ x^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.270 (sec). Leaf size: 57

ode:=diff(diff(x(t),t),t)-x(t)+3*x(t)^2 = 0; 
ic:=[x(0) = 1/4, D(x)(0) = 0]; 
dsolve([ode,op(ic)],x(t), singsol=all);
 

\begin{align*} x &= \operatorname {RootOf}\left (8 \int _{\frac {1}{4}}^{\textit {\_Z}}\frac {1}{\sqrt {-128 \textit {\_a}^{3}+64 \textit {\_a}^{2}-2}}d \textit {\_a} +t \right ) \\ x &= \operatorname {RootOf}\left (8 \int _{\textit {\_Z}}^{\frac {1}{4}}\frac {1}{\sqrt {-128 \textit {\_a}^{3}+64 \textit {\_a}^{2}-2}}d \textit {\_a} +t \right ) \\ \end{align*}

✗ Mathematica

ode=D[x[t],{t,2}]-x[t]+3*x[t]^2==0; 
ic={x[0]==1/4,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(3*x(t)**2 - x(t) + Derivative(x(t), (t, 2)),0) 
ics = {x(0): 1/4, Subs(Derivative(x(t), t), t, 0): 0} 
dsolve(ode,func=x(t),ics=ics)
 

Timed Out