76.3.19 problem 19

Internal problem ID [17319]
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 2.4 (Differences between linear and nonlinear equations). Problems at page 79
Problem number : 19
Date solved : Monday, March 31, 2025 at 03:52:31 PM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 19

ode:=diff(y(t),t) = t*y(t)*(3-y(t)); 
dsolve(ode,y(t), singsol=all);
 

\[ y = \frac {3}{1+3 \,{\mathrm e}^{-\frac {3 t^{2}}{2}} c_1} \]

✓ Mathematica. Time used: 0.272 (sec). Leaf size: 46

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

\begin{align*} y(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-3) K[1]}dK[1]\&\right ]\left [-\frac {t^2}{2}+c_1\right ] \\ y(t)\to 0 \\ y(t)\to 3 \\ \end{align*}

✓ Sympy. Time used: 1.461 (sec). Leaf size: 70

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

\[ \left [ y{\left (t \right )} = \frac {3 \left (- e^{3 t^{2}} - \sqrt {e^{C_{1} + 3 t^{2}}}\right )}{e^{C_{1}} - e^{3 t^{2}}}, \ y{\left (t \right )} = \frac {3 \left (- e^{3 t^{2}} + \sqrt {e^{C_{1} + 3 t^{2}}}\right )}{e^{C_{1}} - e^{3 t^{2}}}\right ] \]