44.5.81 problem 68 (b)

Internal problem ID [7143]
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 (b)
Date solved : Tuesday, February 04, 2025 at 12:38:44 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 )&={\frac {3}{2}} \end{align*}

✓ Solution by Maple

Time used: 18.692 (sec). Leaf size: 1310

dsolve([diff(y(x),x)=(x*(1-x))/(y(x)*(y(x)-2)),y(0) = 3/2],y(x), singsol=all)
 

\begin{align*} \text {Expression too large to display} \\ y &= \frac {2^{{2}/{3}} {\left (\frac {64+\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}+\left (-8 x^{3}+8 \sqrt {\frac {\left (-8-\frac {\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}}{8}+\left (x^{3}-\frac {3}{2} x^{2}-2\right ) \operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )\right ) \left (-8-\frac {\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}}{8}+\left (x^{3}-\frac {3}{2} x^{2}+2\right ) \operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )\right )}{\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}}}+12 x^{2}\right ) \operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )}{\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )}\right )}^{{2}/{3}}+8 \,2^{{1}/{3}}+4 {\left (\frac {64+\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}+\left (-8 x^{3}+8 \sqrt {\frac {\left (-8-\frac {\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}}{8}+\left (x^{3}-\frac {3}{2} x^{2}-2\right ) \operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )\right ) \left (-8-\frac {\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}}{8}+\left (x^{3}-\frac {3}{2} x^{2}+2\right ) \operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )\right )}{\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}}}+12 x^{2}\right ) \operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )}{\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )}\right )}^{{1}/{3}}}{4 {\left (\frac {64+\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}+\left (-8 x^{3}+8 \sqrt {\frac {\left (-8-\frac {\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}}{8}+\left (x^{3}-\frac {3}{2} x^{2}-2\right ) \operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )\right ) \left (-8-\frac {\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}}{8}+\left (x^{3}-\frac {3}{2} x^{2}+2\right ) \operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )\right )}{\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )^{2}}}+12 x^{2}\right ) \operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )}{\operatorname {RootOf}\left (-\textit {\_Z}^{{2}/{3}}+\textit {\_Z}^{{1}/{3}}-4\right )}\right )}^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 4.439 (sec). Leaf size: 242

DSolve[{D[y[x],x]==(x*(1-x))/(y[x]*(y[x]-2)),{y[0]==3/2}},y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {-i 2^{2/3} \sqrt {3} \left (-8 x^3+12 x^2+\sqrt {64 x^6-192 x^5+144 x^4+176 x^3-264 x^2-135}-11\right )^{2/3}-2^{2/3} \left (-8 x^3+12 x^2+\sqrt {64 x^6-192 x^5+144 x^4+176 x^3-264 x^2-135}-11\right )^{2/3}+8 \sqrt [3]{-8 x^3+12 x^2+\sqrt {64 x^6-192 x^5+144 x^4+176 x^3-264 x^2-135}-11}+8 i \sqrt [3]{2} \sqrt {3}-8 \sqrt [3]{2}}{8 \sqrt [3]{-8 x^3+12 x^2+\sqrt {64 x^6-192 x^5+144 x^4+176 x^3-264 x^2-135}-11}} \]