in what coordinates is this equation separable?

Question

In what coordinates are variable separable in the equation:
$$\frac{dy}{dx}=x^2+y^\frac{2}{3}$$ How should I begin working with such these problems?
( I am thinking of changing this to polar coordinates but I do not know what is the exact strategy to solve this)
would you please give me thorough answer to this? thank you.

Answer 1

Hint.

Make the change of variables.

$$ y = \lambda^3(x) x^3 $$

Answer 2

Let $u=y^\frac{1}{3}$ ,

Then $y=u^3$

$\dfrac{dy}{dx}=3u^2\dfrac{du}{dx}$

$\therefore3u^2\dfrac{du}{dx}=x^2+u^2$

$\dfrac{du}{dx}=\dfrac{x^2}{3u^2}+\dfrac{1}{3}$

Let $u=xv$ ,

Then $\dfrac{du}{dx}=x\dfrac{dv}{dx}+v$

$\therefore x\dfrac{dv}{dx}+v=\dfrac{1}{3v^2}+\dfrac{1}{3}$

$x\dfrac{dv}{dx}=\dfrac{1+v^2-3v^3}{3v^2}$

$\dfrac{3v^2}{3v^3-v^2-1}~dv=-\dfrac{dx}{x}$

$\int\dfrac{3v^2}{3v^3-v^2-1}~dv=-\int\dfrac{dx}{x}$