10.4.2 problem 3

Internal problem ID [1183]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Section 2.5. Page 88
Problem number : 3
Date solved : Saturday, March 29, 2025 at 10:44:59 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y \left (-2+y\right ) \left (-1+y\right ) \end{align*}

✓ Maple. Time used: 0.212 (sec). Leaf size: 74

ode:=diff(y(t),t) = y(t)*(-2+y(t))*(-1+y(t)); 
dsolve(ode,y(t), singsol=all);
 

\begin{align*} y &= -\frac {{\mathrm e}^{2 t} c_1}{\left (1+\sqrt {1-{\mathrm e}^{2 t} c_1}\right ) \sqrt {1-{\mathrm e}^{2 t} c_1}} \\ y &= \frac {{\mathrm e}^{2 t} c_1}{\left (1-\sqrt {1-{\mathrm e}^{2 t} c_1}\right ) \sqrt {1-{\mathrm e}^{2 t} c_1}} \\ \end{align*}

✓ Mathematica. Time used: 10.2 (sec). Leaf size: 100

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

\begin{align*} y(t)\to \frac {-\sqrt {1+e^{2 (t+c_1)}}+e^{2 (t+c_1)}+1}{1+e^{2 (t+c_1)}} \\ y(t)\to \frac {\sqrt {1+e^{2 (t+c_1)}}+e^{2 (t+c_1)}+1}{1+e^{2 (t+c_1)}} \\ y(t)\to 0 \\ y(t)\to 1 \\ y(t)\to 2 \\ \end{align*}

✓ Sympy. Time used: 1.347 (sec). Leaf size: 54

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

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