<s>[INST] What is the solution of the equation $xyp^2 + (3x^2 - 2y^2)p - 6xy=0$, where $p = \frac{dx}{dy}$ What is the solution of the equation $xyp^2 + (3x^2 - 2y^2)p - 6xy=0$,  where $p = \frac{dx}{dy}$
I was trying to solve it by dividing the whole equation by $xy$ and then integrate it 
$\frac{dy}{dx}[\frac{dy}{dx} + (\frac{3x}{y} - \frac{2y}{x})] = 6$,  but still this equation is non separable.
Please tell me how to solve it.
 [/INST] Using quadratic formula, we have
$$p=\frac{2y^2 - 3x^2\pm\sqrt{(3x^2-2y^2)^2+24x^2y^2}}{2xy}$$
$$=\frac{2y^2 - 3x^2\pm(3x^2+2y^2)}{2xy}. $$
That is:
$$\frac{dy}{dx}=p=\frac{2y}{x} \text{ or } \frac{-3x}{y}. $$
In both cases, we can easily solve the ODE.
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