57.2.1 problem 1

Internal problem ID [14315]
Book : A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag, NY. 2015.
Section : Chapter 1, First order differential equations. Section 1.1.3 Geometric. Exercises page 15
Problem number : 1
Date solved : Thursday, October 02, 2025 at 09:30:37 AM
CAS classification : [_quadrature]

\begin{align*} x^{\prime }&=x \left (1-\frac {x}{4}\right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 17

ode:=diff(x(t),t) = x(t)*(1-1/4*x(t)); 
dsolve(ode,x(t), singsol=all);
 

\[ x = \frac {4}{1+4 \,{\mathrm e}^{-t} c_1} \]

✓ Mathematica. Time used: 0.159 (sec). Leaf size: 32

ode=D[x[t],t]==x[t]*(1-x[t]/4); 
ic={}; 
DSolve[{ode,ic},x[t],t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to \frac {4 e^t}{e^t+e^{4 c_1}}\\ x(t)&\to 0\\ x(t)&\to 4 \end{align*}

✓ Sympy. Time used: 0.206 (sec). Leaf size: 10

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq((x(t)/4 - 1)*x(t) + Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ x{\left (t \right )} = \frac {4}{C_{1} e^{- t} + 1} \]