65.2.4 problem 7.1 (iv)

Internal problem ID [13638]
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 (iv)
Date solved : Wednesday, March 05, 2025 at 10:05:57 PM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=-x \left (1-x\right ) \left (2-x\right ) \end{align*}

✓ Maple. Time used: 0.155 (sec). Leaf size: 34

ode:=diff(x(t),t) = -x(t)*(-x(t)+1)*(2-x(t)); 
dsolve(ode,x(t), singsol=all);
 

\[ x \left (t \right ) = \frac {c_{1} {\mathrm e}^{t}+\sqrt {-1+{\mathrm e}^{2 t} c_{1}^{2}}}{\sqrt {-1+{\mathrm e}^{2 t} c_{1}^{2}}} \]

✓ Mathematica. Time used: 0.193 (sec). Leaf size: 53

ode=D[x[t],t]==-x[t]*(1-x[t])*(2-x[t]); 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 1.862 (sec). Leaf size: 75

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

\[ \left [ x{\left (t \right )} = \frac {C_{1} - \sqrt {- \left (C_{1} - e^{2 t}\right ) e^{2 t}} - e^{2 t}}{C_{1} - e^{2 t}}, \ x{\left (t \right )} = \frac {\sqrt {- \left (e^{2 C_{1}} - e^{2 t}\right ) e^{2 t}} + e^{2 C_{1}} - e^{2 t}}{e^{2 C_{1}} - e^{2 t}}\right ] \]