76.1.36 problem 36

Internal problem ID [17264]
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 : 36
Date solved : Monday, March 31, 2025 at 03:47:54 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {t y \left (4-y\right )}{1+t} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.060 (sec). Leaf size: 20

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

\[ y = \frac {4}{1+{\mathrm e}^{-4 t} \left (1+t \right )^{4}} \]

✓ Mathematica. Time used: 3.3 (sec). Leaf size: 25

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

\[ y(t)\to \frac {4 e^{4 t}}{(t+1)^4+e^{4 t}} \]

✓ Sympy. Time used: 0.511 (sec). Leaf size: 19

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

\[ y{\left (t \right )} = - \frac {4}{- \left (t + 1\right )^{4} e^{- 4 t} - 1} \]