<s>[INST] Solving the differential equation $\frac{dy}{dx} = \frac{y^3}{2(xy^2-x^2)}$ How can we solve the equation:
$$\frac{dy}{dx} = \frac{y^3}{2(xy^2-x^2)}$$
I get the idea of dividing by $y^2$, But it doesn't become any more solvable (not homogenous).
$$\frac{dy}{dx} = \frac{y}{2(x-\frac{x^2}{y^2})}$$
Substituting $\frac{x}{y} = t$ causes even more complications.
I get an idea of the question to convert into homogenous, but cant form the equation. Please give me a hint!
 [/INST] Nice question! 
Given:
$$\frac{dx}{dy}=\frac{2x}{y}-\frac{2x^2}{y^3}$$
$$=>\frac{dx}{dy}-\frac{2x}{y}=-\frac{2x^2}{y^3}$$
On dividing by $-x^2$,
$$=>\frac{dx}{dy}\frac{(-1)}{x^2}+\frac{2}{xy}=\frac{2}{y^3}$$
Substituting $v=\frac{1}{x}$; $\frac{dv}{dy}=\frac{dx}{dy}\frac{(-1)}{x^2}$gives:
$$\frac{dv}{dy}+\frac{2v}{y}=\frac{2}{y^3}$$
Now simply use the method for solving linear differential equations. 
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