78.6.4 problem 2.c

Internal problem ID [21067]
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.c
Date solved : Thursday, October 02, 2025 at 07:02:41 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y \left (-2+y\right ) \left (3+y\right ) \end{align*}

✓ Maple. Time used: 0.122 (sec). Leaf size: 32

ode:=diff(y(x),x) = y(x)*(y(x)-2)*(3+y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (-2 \textit {\_Z} +30 x +30 c_1 +\ln \left (\frac {\left ({\mathrm e}^{\textit {\_Z}}-3\right )^{5}}{\left ({\mathrm e}^{\textit {\_Z}}-5\right )^{3}}\right )\right )}-3 \]

✓ Mathematica. Time used: 8.084 (sec). Leaf size: 226

ode=D[y[x],x]==y[x]*(y[x]-2)*(y[x]+3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {Root}\left [\text {$\#$1}^5 \left (1+e^{30 x+30 c_1}\right )-15 \text {$\#$1}^3+10 \text {$\#$1}^2+60 \text {$\#$1}-72\&,1\right ]\\ y(x)&\to \text {Root}\left [\text {$\#$1}^5 \left (1+e^{30 x+30 c_1}\right )-15 \text {$\#$1}^3+10 \text {$\#$1}^2+60 \text {$\#$1}-72\&,2\right ]\\ y(x)&\to \text {Root}\left [\text {$\#$1}^5 \left (1+e^{30 x+30 c_1}\right )-15 \text {$\#$1}^3+10 \text {$\#$1}^2+60 \text {$\#$1}-72\&,3\right ]\\ y(x)&\to \text {Root}\left [\text {$\#$1}^5 \left (1+e^{30 x+30 c_1}\right )-15 \text {$\#$1}^3+10 \text {$\#$1}^2+60 \text {$\#$1}-72\&,4\right ]\\ y(x)&\to \text {Root}\left [\text {$\#$1}^5 \left (1+e^{30 x+30 c_1}\right )-15 \text {$\#$1}^3+10 \text {$\#$1}^2+60 \text {$\#$1}-72\&,5\right ]\\ y(x)&\to -3\\ y(x)&\to 0\\ y(x)&\to 2 \end{align*}

✓ Sympy. Time used: 0.869 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2 - y(x))*(y(x) + 3)*y(x) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ x - \frac {\log {\left (y{\left (x \right )} - 2 \right )}}{10} - \frac {\log {\left (y{\left (x \right )} + 3 \right )}}{15} + \frac {\log {\left (y{\left (x \right )} \right )}}{6} = C_{1} \]