Internal problem ID [7886]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 306.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class A], _rational, _dAlembert]
Solve \begin {gather*} \boxed {\left (y^{3}-x^{3}\right ) y^{\prime }-y x^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.985 (sec). Leaf size: 381
dsolve((y(x)^3-x^3)*diff(y(x),x)-x^2*y(x) = 0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {x}{\left (x^{3} c_{1} \left (-x^{3} c_{1}+\sqrt {x^{6} c_{1}^{2}+1}\right )\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {x}{\left (-\left (x^{3} c_{1}+\sqrt {x^{6} c_{1}^{2}+1}\right ) x^{3} c_{1}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {x}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \left (x^{3} c_{1} \left (-x^{3} c_{1}+\sqrt {x^{6} c_{1}^{2}+1}\right )\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \left (x^{3} c_{1} \left (-x^{3} c_{1}+\sqrt {x^{6} c_{1}^{2}+1}\right )\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {x}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \left (x^{3} c_{1} \left (-x^{3} c_{1}+\sqrt {x^{6} c_{1}^{2}+1}\right )\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \left (x^{3} c_{1} \left (-x^{3} c_{1}+\sqrt {x^{6} c_{1}^{2}+1}\right )\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {x}{\left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \left (-\left (x^{3} c_{1}+\sqrt {x^{6} c_{1}^{2}+1}\right ) x^{3} c_{1}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {x}{\left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \left (-\left (x^{3} c_{1}+\sqrt {x^{6} c_{1}^{2}+1}\right ) x^{3} c_{1}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {x}{\left (\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )^{2} \left (-\left (x^{3} c_{1}+\sqrt {x^{6} c_{1}^{2}+1}\right ) x^{3} c_{1}\right )^{\frac {1}{3}}} \\ y \relax (x ) = \frac {x}{\left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )^{2} \left (-\left (x^{3} c_{1}+\sqrt {x^{6} c_{1}^{2}+1}\right ) x^{3} c_{1}\right )^{\frac {1}{3}}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 8.268 (sec). Leaf size: 352
DSolve[-(x^2*y[x]) + (-x^3 + y[x]^3)*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to (-1)^{2/3} \sqrt [3]{x^3-\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to (-1)^{2/3} \sqrt [3]{x^3+\sqrt {x^6-e^{6 c_1}}} \\ y(x)\to 0 \\ y(x)\to \sqrt [3]{x^3-\sqrt {x^6}} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{x^3-\sqrt {x^6}} \\ y(x)\to (-1)^{2/3} \sqrt [3]{x^3-\sqrt {x^6}} \\ y(x)\to \sqrt [3]{\sqrt {x^6}+x^3} \\ y(x)\to -\sqrt [3]{-1} \sqrt [3]{\sqrt {x^6}+x^3} \\ y(x)\to (-1)^{2/3} \sqrt [3]{\sqrt {x^6}+x^3} \\ \end{align*}