Internal problem ID [7879]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, linear first order
Problem number: 299.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, class G], _exact, _rational]
Solve \begin {gather*} \boxed {\left (3 x y^{2}-x^{2}\right ) y^{\prime }+y^{3}-2 y x=0} \end {gather*}
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 327
dsolve((3*x*y(x)^2-x^2)*diff(y(x),x)+y(x)^3-2*x*y(x)=0,y(x), singsol=all)
\begin{align*} y \relax (x ) = \frac {\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}+\frac {2 x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}} \\ y \relax (x ) = -\frac {\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}-\frac {i \sqrt {3}\, \left (\frac {\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {2 x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ y \relax (x ) = -\frac {\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{12 x}-\frac {x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}+\frac {i \sqrt {3}\, \left (\frac {\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}{6 x}-\frac {2 x^{2}}{\left (\left (12 \sqrt {-12 x^{5}+81 c_{1}^{2}}+108 c_{1}\right ) x^{2}\right )^{\frac {1}{3}}}\right )}{2} \\ \end{align*}
✓ Solution by Mathematica
Time used: 1.703 (sec). Leaf size: 288
DSolve[(3*x*y[x]^2-x^2)*y'[x]+y[x]^3-2*x*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {2 \sqrt [3]{3} x^3+\sqrt [3]{2} \left (9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{6^{2/3} x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ y(x)\to \frac {\sqrt [3]{-1} \left (2 \sqrt [3]{3} x^3-\sqrt [3]{-2} \left (9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}\right )}{6^{2/3} x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ y(x)\to \frac {x^3 \text {Root}\left [\text {$\#$1}^3+24\&,2\right ]+\sqrt [3]{-2} \left (9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}\right ){}^{2/3}}{6^{2/3} x \sqrt [3]{9 c_1 x^2+\sqrt {-12 x^9+81 c_1{}^2 x^4}}} \\ \end{align*}