78.6.12 problem 4.c

Internal problem ID [21075]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 7, Nonlinear systems. Problems section 7.11
Problem number : 4.c
Date solved : Thursday, October 02, 2025 at 07:04:33 PM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=\mu -x^{3} \end{align*}

✓ Maple. Time used: 0.205 (sec). Leaf size: 83

ode:=diff(x(t),t) = mu-x(t)^3; 
dsolve(ode,x(t), singsol=all);
 

\[ x = \frac {\mu ^{{1}/{3}} \sqrt {3}\, \left (-\sqrt {3}+3 \tan \left (\operatorname {RootOf}\left (6 \sqrt {3}\, c_1 \,\mu ^{{2}/{3}}+6 \sqrt {3}\, t \,\mu ^{{2}/{3}}+2 \sqrt {3}\, \ln \left (\mu ^{{1}/{3}} \left (-3+\sqrt {3}\, \tan \left (\textit {\_Z} \right )\right )\right )-\sqrt {3}\, \ln \left (\mu ^{{2}/{3}} \sec \left (\textit {\_Z} \right )^{2}\right )-\sqrt {3}\, \ln \left (3\right )-6 \textit {\_Z} \right )\right )\right )}{6} \]

✓ Mathematica. Time used: 0.259 (sec). Leaf size: 120

ode=D[x[t],t]==\[Mu]-x[t]^3; 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \text {InverseFunction}\left [-\frac {\log \left (\text {$\#$1}^2+\text {$\#$1} \sqrt [3]{\mu }+\mu ^{2/3}\right )+2 \sqrt {3} \arctan \left (\frac {\frac {2 \text {$\#$1}}{\sqrt [3]{\mu }}+1}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{\mu }-\text {$\#$1}\right )}{6 \mu ^{2/3}}\&\right ][-t+c_1]\\ x(t)&\to \sqrt [3]{\mu }\\ x(t)&\to -\sqrt [3]{-1} \sqrt [3]{\mu }\\ x(t)&\to (-1)^{2/3} \sqrt [3]{\mu } \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 24

from sympy import * 
t = symbols("t") 
mu = symbols("mu") 
x = Function("x") 
ode = Eq(-mu + x(t)**3 + Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ \operatorname {RootSum} {\left (27 t^{3} \mu ^{2} - 1, \left ( t \mapsto t \log {\left (- 3 t \mu + x{\left (t \right )} \right )} \right )\right )} = C_{1} - t \]