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 : Tuesday, February 04, 2025 at 12:39:32 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*}

✓ Solution by Maple

Time used: 0.266 (sec). Leaf size: 158

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

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

✓ Solution by Mathematica

Time used: 3.500 (sec). Leaf size: 242

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

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