44.5.82 problem 68 (c)

Internal problem ID [7144]
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 2.2 Separable equations. Exercises 2.2 at page 53
Problem number : 68 (c)
Date solved : Sunday, March 30, 2025 at 11:48:51 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=-2 \end{align*}

✓ Maple. Time used: 0.350 (sec). Leaf size: 158

ode:=diff(y(x),x) = x*(1-x)/y(x)/(y(x)-2); 
ic:=y(0) = -2; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {\left (i \sqrt {3}-1\right ) \left (-72-4 x^{3}+6 x^{2}+2 \sqrt {4 x^{6}-12 x^{5}+9 x^{4}+144 x^{3}-216 x^{2}+1280}\right )^{{2}/{3}}-4 i \sqrt {3}+4 \left (-72-4 x^{3}+6 x^{2}+2 \sqrt {4 x^{6}-12 x^{5}+9 x^{4}+144 x^{3}-216 x^{2}+1280}\right )^{{1}/{3}}-4}{4 \left (-72-4 x^{3}+6 x^{2}+2 \sqrt {4 x^{6}-12 x^{5}+9 x^{4}+144 x^{3}-216 x^{2}+1280}\right )^{{1}/{3}}} \]

✓ Mathematica. Time used: 3.5 (sec). Leaf size: 242

ode=D[y[x],x]==(x*(1-x))/(y[x]*(y[x]-2)); 
ic={y[0]==-2}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {i \sqrt [3]{2} \sqrt {3} \left (-2 x^3+3 x^2+\sqrt {4 x^6-12 x^5+9 x^4+144 x^3-216 x^2+1280}-36\right )^{2/3}-\sqrt [3]{2} \left (-2 x^3+3 x^2+\sqrt {4 x^6-12 x^5+9 x^4+144 x^3-216 x^2+1280}-36\right )^{2/3}+4 \sqrt [3]{-2 x^3+3 x^2+\sqrt {4 x^6-12 x^5+9 x^4+144 x^3-216 x^2+1280}-36}-2 i 2^{2/3} \sqrt {3}-2\ 2^{2/3}}{4 \sqrt [3]{-2 x^3+3 x^2+\sqrt {4 x^6-12 x^5+9 x^4+144 x^3-216 x^2+1280}-36}} \]

✗ Sympy

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

ZeroDivisionError : polynomial division