63.2.1 problem 1

Internal problem ID [13034]
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 : Tuesday, January 28, 2025 at 04:49:49 AM
CAS classification : [_quadrature]

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

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

Time used: 0.230 (sec). Leaf size: 44

DSolve[D[x[t],t]==x[t]*(1-x[t]/4),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*}