To solve homogeneous first order ODEs of the form $$\frac{dy}{dx} = \frac{f(x,y)}{g(x, y)} \tag 1 $$ the method seems to be to substitute $y=vx$, take an $x^n$ out of both numerator and denominator in $(1),$ cancel them out, and to proceed.
Wouldn't there be a loss of the root $x=0$ in this method? If so, why's it used?
EDIT: This seems to be a common problem in differential equations. Loss of roots is a pretty well-known thing, but I'm seeing multiple cases in practice questions here where both differentials and variables are blatantly divided across equations without a care. Is there any reason for this?
After substitution you get $$ xv'+v=\frac{f(1,v)}{g(1,v)}. $$ This equation is singular in $x=0$, so the question of cancellation of factors $x$ is meaningless.
After solving for $x\ne 0$ one can examine if the solution $y(x)=xv(x)$ extends to $x=0$ as a continuous function or even as ODE solution.