87.3.6 problem 6

Internal problem ID [23255]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 26
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:27:10 PM
CAS classification : [_quadrature]

\begin{align*} p^{\prime }&=a p-b p^{2} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 18

ode:=diff(p(t),t) = a*p(t)-b*p(t)^2; 
dsolve(ode,p(t), singsol=all);
 

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

✓ Mathematica. Time used: 0.395 (sec). Leaf size: 43

ode=D[p[t],t]==a*p[t]-b*p[t]^2; 
ic={}; 
DSolve[{ode,ic},p[t],t,IncludeSingularSolutions->True]
 

\begin{align*} p(t)&\to \frac {a e^{a (t+c_1)}}{1+b e^{a (t+c_1)}}\\ p(t)&\to 0\\ p(t)&\to \frac {a}{b} \end{align*}

✓ Sympy. Time used: 0.252 (sec). Leaf size: 44

from sympy import * 
t = symbols("t") 
a = symbols("a") 
b = symbols("b") 
p = Function("p") 
ode = Eq(-a*p(t) + b*p(t)**2 + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=p(t),ics=ics)
 

\[ \left [ p{\left (t \right )} = \frac {a - \sqrt {a^{2} - 4 b \frac {d}{d x} y{\left (x \right )}}}{2 b}, \ p{\left (t \right )} = \frac {a + \sqrt {a^{2} - 4 b \frac {d}{d x} y{\left (x \right )}}}{2 b}\right ] \]