63.2.1 problem 1

Internal problem ID [12955]
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 : Wednesday, March 05, 2025 at 08:54:48 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 17

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

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

✓ Mathematica. Time used: 0.23 (sec). Leaf size: 44

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 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-4) K[1]}dK[1]\&\right ]\left [-\frac {t}{4}+c_1\right ] \\ x(t)\to 0 \\ x(t)\to 4 \\ \end{align*}

✓ Sympy. Time used: 0.386 (sec). Leaf size: 10

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

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