Differential equation, hint

Question

$$x^2y'^2-2(xy-2)y' +y^2=0$$

I have tried to determine y' using x, y

$$y'=-\frac{{\sqrt{1-xy}-1}^2}{x^2}$$

And I don't know what to do next.

Answer 1

As a good start, let $y = vx$. The equation in terms of $v$ and $x$ looks a lot simpler; something like $$x^4 v'^2 + 4x v' +4v = 0$$ Then a substitution of a form like $v = w + 2 x^{-3}$ gives something like $$ w'^2 = f(x) $$ which you can solve the way you started trying to solve the original.