1. ## Tricky homogenous equation

$xy^2 y' = y^3 + x^2 \sqrt{x^2 - y^2}$

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

I know that everything on the right side of the equation needs to be in terms of y/x, but I can't find a way to make this happen.

2. $\frac{x\sqrt{x^2 - y^2}}{y^2} = \frac{x}{y}\left(\frac{\sqrt{x^2 - y^2}}{y}\right)$

$= \frac{1}{\frac{y}{x}}\left(\frac{\sqrt{x^2 - y^2}}{\sqrt{y^2}}\right)$

$= \frac{1}{\frac{y}{x}}\left(\sqrt{\frac{x^2 - y^2}{y^2}}\right)$

$= \frac{1}{\frac{y}{x}}\left[\sqrt{\left(\frac{x}{y}\right)^2 - 1}\right]$.

Now make the substitution $v = \frac{y}{x}$, that means $y = vx$ and $\frac{dy}{dx} = x\,\frac{dv}{dx} + v$.