81.17.5 problem 21-51

Internal problem ID [21738]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 21. Applications of second order differential equations
Problem number : 21-51
Date solved : Thursday, October 02, 2025 at 08:01:37 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=\alpha \left (A -y\right ) y \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 19

ode:=diff(y(x),x) = alpha*(A-y(x))*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {A}{1+{\mathrm e}^{-\alpha A x} c_1 A} \]

✓ Mathematica. Time used: 0.407 (sec). Leaf size: 41

ode=D[y[x],x]==\[Alpha]*(A-y[x])*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {A e^{A (\alpha x+c_1)}}{-1+e^{A (\alpha x+c_1)}}\\ y(x)&\to 0\\ y(x)&\to A \end{align*}

✓ Sympy. Time used: 0.237 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = \frac {A e^{A \left (C_{1} + a x\right )}}{e^{A \left (C_{1} + a x\right )} - 1} \]