38.4.38 problem 20

Internal problem ID [8337]
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 : 20
Date solved : Tuesday, September 30, 2025 at 05:29:10 PM
CAS classification : [_quadrature]

\begin{align*} y^{\prime }&=y^{2}-y^{4} \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 47

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

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (\ln \left ({\mathrm e}^{\textit {\_Z}}-2\right ) {\mathrm e}^{\textit {\_Z}}+2 c_1 \,{\mathrm e}^{\textit {\_Z}}-\textit {\_Z} \,{\mathrm e}^{\textit {\_Z}}+2 x \,{\mathrm e}^{\textit {\_Z}}-\ln \left ({\mathrm e}^{\textit {\_Z}}-2\right )-2 c_1 +\textit {\_Z} -2 x +2\right )}-1 \]

✓ Mathematica. Time used: 0.114 (sec). Leaf size: 53

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{(K[1]-1) K[1]^2 (K[1]+1)}dK[1]\&\right ][-x+c_1]\\ y(x)&\to -1\\ y(x)&\to 0\\ y(x)&\to 1 \end{align*}

✓ Sympy. Time used: 0.167 (sec). Leaf size: 24

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

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