14.29.11 problem 11

Internal problem ID [2809]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.2 (Stability of linear systems). Page 383
Problem number : 11
Date solved : Sunday, March 30, 2025 at 12:20:57 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 14

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

\[ x = \frac {1}{1+{\mathrm e}^{-t} c_1} \]

✓ Mathematica. Time used: 0.309 (sec). Leaf size: 29

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

\begin{align*} x(t)\to \frac {e^t}{e^t+e^{c_1}} \\ x(t)\to 0 \\ x(t)\to 1 \\ \end{align*}

✓ Sympy. Time used: 0.332 (sec). Leaf size: 10

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

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