<s>[INST] Struggles solving : $\frac23xyy'=\sqrt{x^6-y^4}+y^2.$ 
Solve $\frac23xyy'=\sqrt{x^6-y^4}+y^2.$
(Hint: the equation can be boiled down to a homogeneous one.)

My attempt:
I didn't know how to obtain a homogeneous equation, so I'm going to show what I did instead.
We can write $\frac23xyy'=|x|^3\sqrt{1-\frac{y^4}{x^6}}+y^2.$ Let $z=\frac{y^2}{x^3}=y^2x^{-3}.$ Then
$$\begin{aligned}z'&=2yy'x^{-3}-3y^2x^{-4}\\ \implies x^4z'&=2xyy'-3y^2\\ \implies\frac23xyy'&=\frac{x^4z'+3y^2}3=\frac{x^4z'}3+y^2\\ \implies\frac{x^4z'}3+y^2&=|x|^3\sqrt{1-z^2}+y^2\\ \implies\frac{|x|z'}3&=\sqrt{1-z^2}\\ \implies\frac{dz}{\sqrt{1-z^2}}&=3\frac{dx}{|x|}\\ \implies\arcsin z&=3\ln|x|+C, C\in\Bbb R\\ \implies z&=\sin\ln(C|x|^3), C>0\\ \implies y^2&=x^3\sin\ln(C|x|^3)C>0.\end{aligned}$$
I'm not sure my answer is correct. I didn't get far plugging the result into the equation.
How do we solve the given ODE? Or should I better ask, how do we obtain a homogeneous ODE from it?
 [/INST] $$\frac23xyy'=\sqrt{x^6-y^4}+y^2.$$
$$x^3(y^2)'=3x^2\sqrt{x^6-y^4}+3x^2y^2$$
$$\dfrac {x^3(y^2)'-3x^2y^2}{x^6}=\dfrac 3 {x^4}\sqrt{x^6-y^4}$$
$$\left (\dfrac {y^2}{x^3}\right)'=\dfrac 3 {x^4}\sqrt{x^6-y^4}$$
$$\left (\dfrac {y^2}{x^3}\right)'=\dfrac 3 {x}\sqrt{1-\left (\dfrac {y^2}{x^3}\right)^2}$$
The DE is separable.
Substitute $u=\left (\dfrac {y^2}{x^3}\right)$:
$$\dfrac {du}{ \sqrt{1- u^2}}=\dfrac 3x dx$$
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