59.2.5 problem 7.1 (v)

Internal problem ID [14999]
Book : AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section : Chapter 7, Scalar autonomous ODEs. Exercises page 56
Problem number : 7.1 (v)
Date solved : Thursday, October 02, 2025 at 09:58:16 AM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=x^{2}-x^{4} \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 47

ode:=diff(x(t),t) = x(t)^2-x(t)^4; 
dsolve(ode,x(t), singsol=all);
 

\[ x = {\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}-2\right ) {\mathrm e}^{\textit {\_Z}}+2 c_1 \,{\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 t \,{\mathrm e}^{\textit {\_Z}}-\ln \left ({\mathrm e}^{\textit {\_Z}}-2\right )-2 c_1 +\textit {\_Z} -2 t +2\right )}-1 \]

✓ Mathematica. Time used: 0.166 (sec). Leaf size: 53

ode=D[x[t],t]==x[t]^2-x[t]^4; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \text {InverseFunction}\left [\frac {1}{\text {$\#$1}}+\frac {1}{2} \log (1-\text {$\#$1})-\frac {1}{2} \log (\text {$\#$1}+1)\&\right ][-t+c_1]\\ x(t)&\to -1\\ x(t)&\to 0\\ x(t)&\to 1 \end{align*}

✓ Sympy. Time used: 0.182 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(x(t)**4 - x(t)**2 + Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ t + \frac {\log {\left (x{\left (t \right )} - 1 \right )}}{2} - \frac {\log {\left (x{\left (t \right )} + 1 \right )}}{2} + \frac {1}{x{\left (t \right )}} = C_{1} \]