44.4.43 problem 25

Internal problem ID [7056]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 2. First order differential equations. Section 2.1 Solution curves without a solution. Exercises 2.1 at page 44
Problem number : 25
Date solved : Sunday, March 30, 2025 at 11:37:02 AM
CAS classification : [_quadrature]

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

✓ Maple. Time used: 0.028 (sec). Leaf size: 49

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

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}-4\right ) {\mathrm e}^{\textit {\_Z}}+16 c_1 \,{\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+16 x \,{\mathrm e}^{\textit {\_Z}}-2 \ln \left ({\mathrm e}^{\textit {\_Z}}-4\right )-32 c_1 +2 \textit {\_Z} -32 x +4\right )}-2 \]

✓ Mathematica. Time used: 0.237 (sec). Leaf size: 57

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

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {1}{4 \text {$\#$1}}+\frac {1}{16} \log (2-\text {$\#$1})-\frac {1}{16} \log (\text {$\#$1}+2)\&\right ][-x+c_1] \\ y(x)\to -2 \\ y(x)\to 0 \\ y(x)\to 2 \\ \end{align*}

✓ Sympy. Time used: 0.440 (sec). Leaf size: 26

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

\[ x + \frac {\log {\left (y{\left (x \right )} - 2 \right )}}{16} - \frac {\log {\left (y{\left (x \right )} + 2 \right )}}{16} + \frac {1}{4 y{\left (x \right )}} = C_{1} \]