78.6.11 problem 4.b

Internal problem ID [21074]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS FOR SCIENTISTS AND ENGINEERS. By Russell Herman. University of North Carolina Wilmington. LibreText. compiled on 06/09/2025
Section : Chapter 7, Nonlinear systems. Problems section 7.11
Problem number : 4.b
Date solved : Thursday, October 02, 2025 at 07:03:21 PM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 73

ode:=diff(y(x),x) = y(x)*(mu-y(x))*(mu-2*y(x)); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {\left (\sqrt {-4 \,{\mathrm e}^{\left (c_1 +x \right ) \mu ^{2}}+1}-1\right ) \mu }{2 \sqrt {-4 \,{\mathrm e}^{\left (c_1 +x \right ) \mu ^{2}}+1}} \\ y &= \frac {\left (\sqrt {-4 \,{\mathrm e}^{\left (c_1 +x \right ) \mu ^{2}}+1}+1\right ) \mu }{2 \sqrt {-4 \,{\mathrm e}^{\left (c_1 +x \right ) \mu ^{2}}+1}} \\ \end{align*}

✓ Mathematica. Time used: 0.252 (sec). Leaf size: 59

ode=D[y[x],x]==y[x]*(\[Mu]*y[x])*(\[Mu]-2*y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [-\frac {2 \log (\text {$\#$1})}{\mu ^2}+\frac {2 \log (2 \text {$\#$1}-\mu )}{\mu ^2}+\frac {1}{\text {$\#$1} \mu }\&\right ][-\mu x+c_1]\\ y(x)&\to 0\\ y(x)&\to \frac {\mu }{2} \end{align*}

✓ Sympy. Time used: 1.132 (sec). Leaf size: 44

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

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