65.2.5 problem 7.1 (v)

Internal problem ID [13718]
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 : Tuesday, January 28, 2025 at 05:59:35 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

Time used: 0.166 (sec). Leaf size: 47

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

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

✓ Solution by Mathematica

Time used: 0.206 (sec). Leaf size: 53

DSolve[D[x[t],t]==x[t]^2-x[t]^4,x[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]^2 (K[1]+1)}dK[1]\&\right ][-t+c_1] \\ x(t)\to -1 \\ x(t)\to 0 \\ x(t)\to 1 \\ \end{align*}