72.2.7 problem 7

Internal problem ID [14563]
Book : DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section : Chapter 1. First-Order Differential Equations. Exercises section 1.3 page 47
Problem number : 7
Date solved : Thursday, March 13, 2025 at 03:34:18 AM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=3 y \left (1-y\right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{2}} \end{align*}

✓ Maple. Time used: 0.018 (sec). Leaf size: 12

ode:=diff(y(t),t) = 3*y(t)*(1-y(t)); 
ic:=y(0) = 1/2; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {1}{1+{\mathrm e}^{-3 t}} \]

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 20

ode=D[y[t],t]==3*y[t]*(1-y[t]); 
ic={y[0]==1/2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {e^{3 t}}{e^{3 t}+1} \]

✓ Sympy. Time used: 0.377 (sec). Leaf size: 12

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-3*(1 - y(t))*y(t) + Derivative(y(t), t),0) 
ics = {y(0): 1/2} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {1}{1 + e^{- 3 t}} \]