76.1.35 problem 35

Internal problem ID [17263]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.1 (Separable equations). Problems at page 44
Problem number : 35
Date solved : Monday, March 31, 2025 at 03:47:51 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {t y \left (4-y\right )}{3} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.053 (sec). Leaf size: 18

ode:=diff(y(t),t) = 1/3*t*y(t)*(4-y(t)); 
ic:=y(0) = 1/2; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = \frac {4}{1+7 \,{\mathrm e}^{-\frac {2 t^{2}}{3}}} \]

✓ Mathematica. Time used: 0.26 (sec). Leaf size: 29

ode=D[y[t],t]==t*y[t]*(4-y[t])/3; 
ic={y[0]==1/2}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ y(t)\to \frac {4 e^{\frac {2 t^2}{3}}}{e^{\frac {2 t^2}{3}}+7} \]

✗ Sympy

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t*(4 - y(t))*y(t)/3 + Derivative(y(t), t),0) 
ics = {y(0): 1/2} 
dsolve(ode,func=y(t),ics=ics)
 

Timed Out