78.6.5 problem 2.d

Internal problem ID [21068]
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 : 2.d
Date solved : Thursday, October 02, 2025 at 07:02:42 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y^{2} \left (1+y\right ) \left (y-4\right ) \end{align*}

✓ Maple. Time used: 0.181 (sec). Leaf size: 62

ode:=diff(y(x),x) = y(x)^2*(1+y(x))*(y(x)-4); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (-\ln \left (\left ({\mathrm e}^{\textit {\_Z}}-5\right ) \left ({\mathrm e}^{\textit {\_Z}}-1\right )^{15}\right ) {\mathrm e}^{\textit {\_Z}}+80 c_1 \,{\mathrm e}^{\textit {\_Z}}+16 \textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+80 x \,{\mathrm e}^{\textit {\_Z}}+\ln \left (\left ({\mathrm e}^{\textit {\_Z}}-5\right ) \left ({\mathrm e}^{\textit {\_Z}}-1\right )^{15}\right )-80 c_1 -16 \textit {\_Z} -80 x -20\right )}-1 \]

✓ Mathematica. Time used: 40.251 (sec). Leaf size: 62

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {1}{4 \text {$\#$1}}+\frac {1}{80} \log (4-\text {$\#$1})+\frac {3 \log (\text {$\#$1})}{16}-\frac {1}{5} \log (\text {$\#$1}+1)\&\right ][x+c_1]\\ y(x)&\to -1\\ y(x)&\to 0\\ y(x)&\to 4 \end{align*}

✓ Sympy. Time used: 0.294 (sec). Leaf size: 34

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

\[ x - \frac {\log {\left (y{\left (x \right )} - 4 \right )}}{80} + \frac {\log {\left (y{\left (x \right )} + 1 \right )}}{5} - \frac {3 \log {\left (y{\left (x \right )} \right )}}{16} - \frac {1}{4 y{\left (x \right )}} = C_{1} \]