38.1.21 problem 23

Internal problem ID [8182]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 1. Introduction to differential equations. Exercises 1.1 at page 12
Problem number : 23
Date solved : Tuesday, September 30, 2025 at 05:18:23 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 14

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

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

✓ Mathematica. Time used: 0.12 (sec). Leaf size: 42

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

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

✓ Sympy. Time used: 0.210 (sec). Leaf size: 10

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

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