ordinary differential equations - Separable ODE's

I came across this question in my tutorial sheet for my ODE class, and got a bit stuck.

The differential equation $x y^\prime =y ( \ln (x)- \ln (y))$ is neither separable, nor linear. By making the substitution $y(x) = xv(x)$, show that the new equation for $v(x)$ is a separable equation.

My working:
substitute $y(x) = x v(x)$ so,

$x y^\prime = x v( \ln(x)- \ln(y))$

$y^\prime =v(\ln(x)-\ln(y))$

Now, the part Im confused about is how to actually show that this equation is separable. Any help would be much appreciated, thanks in advance.

Answer

We have

$$y = v x \implies y' = v + v' x$$

Substituting (notice you did not substitute $y'$ in order to reduce it to a Separable Equation).

$$x y' = x (v + v' x) = y(\ln(x) - \ln(y)) = v x(\ln(x) - \ln(vx))$$

Can you continue now?

Hint: Start by solving for $v'$ and then see what you have.