Solve $(2x^3+3x^2y+y^2-y^3)dx+(2y^3+3xy^2+x^2-x^3)dy=0$ 
Solve the differential equation
$$(2x^3+3x^2y+y^2-y^3)dx+(2y^3+3xy^2+x^2-x^3)dy=0$$

I don't know how to solve it, and I have tried many methods,but they didn't work!

Hint: Let:
$$y = v x \rightarrow \dfrac{dy}{dx} = v + x \dfrac{dv}{dx}$$
Substitute this into
$$(2x^3+3x^2y+y^2-y^3)+(2y^3+3xy^2+x^2-x^3)~\dfrac{dy}{dx} = 0$$
It looks nasty!
Anyway, you will get a solvable equation after you use an integrating factor to make it an exact equation and end up with:
$v(x) = \dfrac{\sqrt[3]{2} \left(c_1-x\right)}{\sqrt[3]{27 c_1 x^4+\sqrt{108 x^6 \left(x-c_1\right){}^3+729 \left(x^6-c_1 x^4\right){}^2}-27 x^6}}+\dfrac{\sqrt[3]{27 c_1 x^4+\sqrt{108 x^6 \left(x-c_1\right){}^3+729 \left(x^6-c_1 x^4\right){}^2}-27 x^6}}{3 \sqrt[3]{2} x^2}$
From the original substitution of $y = v x$, eliminate $v$ in the above and solve for $y$.
Final solution:
$y(x)=\dfrac{\sqrt[3]{2} \left(3 c_1+3 x\right)}{3 \sqrt[3]{\sqrt{\left(27 c_1 x+27 x^3\right){}^2+4 \left(3 c_1+3 x\right){}^3}+27 c_1 x+27 x^3}}-\dfrac{\sqrt[3]{\sqrt{\left(27 c_1 x+27 x^3\right){}^2+4 \left(3 c_1+3 x\right){}^3}+27 c_1 x+27 x^3}}{3 \sqrt[3]{2}}$
