78.8.5 problem 5

Internal problem ID [18203]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 2. First order equations. Miscellaneous Problems for Chapter 2. Problems at page 99
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 11:38:13 AM
CAS classification : [[_homogeneous, `class G`], _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} y^{2}&=\left (x^{3}-y x \right ) y^{\prime } \end{align*}

✓ Solution by Maple

Time used: 0.119 (sec). Leaf size: 207

dsolve(y(x)^2=(x^3-x*y(x))*diff(y(x),x),y(x), singsol=all)
 

\begin{align*} y &= \frac {c_{1} \left (\left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{{2}/{3}}+c_{1} \right )}{x \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{{1}/{3}}} \\ y &= -\frac {c_{1} \left (i \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{{2}/{3}} \sqrt {3}-i \sqrt {3}\, c_{1} +\left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{{2}/{3}}+c_{1} \right )}{2 x \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{{1}/{3}}} \\ y &= -\frac {c_{1} \left (-i \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{{2}/{3}} \sqrt {3}+i \sqrt {3}\, c_{1} +\left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{{2}/{3}}+c_{1} \right )}{2 x \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 60.115 (sec). Leaf size: 820

DSolve[y[x]^2==(x^3-x*y[x])*D[y[x],x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to x^2-\frac {9 x^2}{\frac {9 \sqrt [3]{x^{12} \left (-\cosh \left (\frac {3 c_1}{4}\right )\right )-x^{12} \sinh \left (\frac {3 c_1}{4}\right )+2 x^6 \cosh \left (\frac {3 c_1}{8}\right )+2 x^6 \sinh \left (\frac {3 c_1}{8}\right )+\sqrt {x^6 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right ) \left (\left (x^6-1\right ) \cosh \left (\frac {3 c_1}{16}\right )+\left (x^6+1\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3}-1}}{x^6 \cosh \left (\frac {3 c_1}{8}\right )+x^6 \sinh \left (\frac {3 c_1}{8}\right )-1}-\frac {9}{\sqrt [3]{x^{12} \left (-\cosh \left (\frac {3 c_1}{4}\right )\right )-x^{12} \sinh \left (\frac {3 c_1}{4}\right )+2 x^6 \cosh \left (\frac {3 c_1}{8}\right )+2 x^6 \sinh \left (\frac {3 c_1}{8}\right )+\sqrt {x^6 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right ) \left (\left (x^6-1\right ) \cosh \left (\frac {3 c_1}{16}\right )+\left (x^6+1\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3}-1}}+9} \\ y(x)\to x^2-\frac {18 x^2}{\frac {9 i \left (\sqrt {3}+i\right ) \sqrt [3]{x^{12} \left (-\cosh \left (\frac {3 c_1}{4}\right )\right )-x^{12} \sinh \left (\frac {3 c_1}{4}\right )+2 x^6 \cosh \left (\frac {3 c_1}{8}\right )+2 x^6 \sinh \left (\frac {3 c_1}{8}\right )+\sqrt {x^6 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right ) \left (\left (x^6-1\right ) \cosh \left (\frac {3 c_1}{16}\right )+\left (x^6+1\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3}-1}}{x^6 \cosh \left (\frac {3 c_1}{8}\right )+x^6 \sinh \left (\frac {3 c_1}{8}\right )-1}+\frac {9+9 i \sqrt {3}}{\sqrt [3]{x^{12} \left (-\cosh \left (\frac {3 c_1}{4}\right )\right )-x^{12} \sinh \left (\frac {3 c_1}{4}\right )+2 x^6 \cosh \left (\frac {3 c_1}{8}\right )+2 x^6 \sinh \left (\frac {3 c_1}{8}\right )+\sqrt {x^6 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right ) \left (\left (x^6-1\right ) \cosh \left (\frac {3 c_1}{16}\right )+\left (x^6+1\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3}-1}}+18} \\ y(x)\to x^2-\frac {18 x^2}{-\frac {9 i \left (\sqrt {3}-i\right ) \sqrt [3]{x^{12} \left (-\cosh \left (\frac {3 c_1}{4}\right )\right )-x^{12} \sinh \left (\frac {3 c_1}{4}\right )+2 x^6 \cosh \left (\frac {3 c_1}{8}\right )+2 x^6 \sinh \left (\frac {3 c_1}{8}\right )+\sqrt {x^6 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right ) \left (\left (x^6-1\right ) \cosh \left (\frac {3 c_1}{16}\right )+\left (x^6+1\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3}-1}}{x^6 \cosh \left (\frac {3 c_1}{8}\right )+x^6 \sinh \left (\frac {3 c_1}{8}\right )-1}+\frac {9-9 i \sqrt {3}}{\sqrt [3]{x^{12} \left (-\cosh \left (\frac {3 c_1}{4}\right )\right )-x^{12} \sinh \left (\frac {3 c_1}{4}\right )+2 x^6 \cosh \left (\frac {3 c_1}{8}\right )+2 x^6 \sinh \left (\frac {3 c_1}{8}\right )+\sqrt {x^6 \left (\cosh \left (\frac {15 c_1}{16}\right )+\sinh \left (\frac {15 c_1}{16}\right )\right ) \left (\left (x^6-1\right ) \cosh \left (\frac {3 c_1}{16}\right )+\left (x^6+1\right ) \sinh \left (\frac {3 c_1}{16}\right )\right ){}^3}-1}}+18} \\ \end{align*}