Internal problem ID [2608]
Book: Differential equations with applications and historial notes, George F. Simmons,
1971
Section: Chapter 2, End of chapter, page 61
Problem number: 5.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _rational, [_Abel, 2nd type, class B]]
Solve \begin {gather*} \boxed {y^{2}-\left (x^{3}-y x \right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.302 (sec). Leaf size: 285
dsolve(y(x)^2=(x^3-x*y(x))*diff(y(x),x),y(x), singsol=all)
\begin{align*} y \relax (x ) = c_{1} \left (\frac {\left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{\frac {1}{3}}}{x^{3}}+\frac {c_{1}}{x^{3} \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{\frac {1}{3}}}\right ) x^{2} \\ y \relax (x ) = \frac {c_{1} \left (-\frac {2 \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{\frac {1}{3}}}{x^{3}}-\frac {2 c_{1}}{x^{3} \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{\frac {1}{3}}}-2 i \sqrt {3}\, \left (\frac {\left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{\frac {1}{3}}}{x^{3}}-\frac {c_{1}}{x^{3} \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{\frac {1}{3}}}\right )\right ) x^{2}}{4} \\ y \relax (x ) = \frac {c_{1} \left (-\frac {2 \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{\frac {1}{3}}}{x^{3}}-\frac {2 c_{1}}{x^{3} \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{\frac {1}{3}}}+2 i \sqrt {3}\, \left (\frac {\left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{\frac {1}{3}}}{x^{3}}-\frac {c_{1}}{x^{3} \left (-x^{3}+\sqrt {x^{6}-c_{1}^{3}}\right )^{\frac {1}{3}}}\right )\right ) x^{2}}{4} \\ \end{align*}
✓ Solution by Mathematica
Time used: 95.156 (sec). Leaf size: 550
DSolve[y[x]^2==(x^3-x*y[x])*y'[x],y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to x^2 \left (1-\frac {9}{\frac {9 \sqrt [3]{-e^{\frac {3 c_1}{4}} x^{12}+2 e^{\frac {3 c_1}{8}} x^6+\sqrt {e^{\frac {3 c_1}{8}} x^6 \left (-1+e^{\frac {3 c_1}{8}} x^6\right ){}^3}-1}}{-1+e^{\frac {3 c_1}{8}} x^6}-\frac {9}{\sqrt [3]{-e^{\frac {3 c_1}{4}} x^{12}+2 e^{\frac {3 c_1}{8}} x^6+\sqrt {e^{\frac {3 c_1}{8}} x^6 \left (-1+e^{\frac {3 c_1}{8}} x^6\right ){}^3}-1}}+9}\right ) \\ y(x)\to x^2 \left (1-\frac {18}{\frac {9 i \left (\sqrt {3}+i\right ) \sqrt [3]{-e^{\frac {3 c_1}{4}} x^{12}+2 e^{\frac {3 c_1}{8}} x^6+\sqrt {e^{\frac {3 c_1}{8}} x^6 \left (-1+e^{\frac {3 c_1}{8}} x^6\right ){}^3}-1}}{-1+e^{\frac {3 c_1}{8}} x^6}+\frac {9+9 i \sqrt {3}}{\sqrt [3]{-e^{\frac {3 c_1}{4}} x^{12}+2 e^{\frac {3 c_1}{8}} x^6+\sqrt {e^{\frac {3 c_1}{8}} x^6 \left (-1+e^{\frac {3 c_1}{8}} x^6\right ){}^3}-1}}+18}\right ) \\ y(x)\to x^2 \left (1-\frac {18}{\frac {\left (-9-9 i \sqrt {3}\right ) \sqrt [3]{-e^{\frac {3 c_1}{4}} x^{12}+2 e^{\frac {3 c_1}{8}} x^6+\sqrt {e^{\frac {3 c_1}{8}} x^6 \left (-1+e^{\frac {3 c_1}{8}} x^6\right ){}^3}-1}}{-1+e^{\frac {3 c_1}{8}} x^6}+\frac {9-9 i \sqrt {3}}{\sqrt [3]{-e^{\frac {3 c_1}{4}} x^{12}+2 e^{\frac {3 c_1}{8}} x^6+\sqrt {e^{\frac {3 c_1}{8}} x^6 \left (-1+e^{\frac {3 c_1}{8}} x^6\right ){}^3}-1}}+18}\right ) \\ y(x)\to 0 \\ y(x)\to \frac {2 x^2}{3} \\ \end{align*}