Why is it legitimate to solve the differential equation $frac{dy}{dx}=frac{y}{x}$ by taking $int frac{1}{y} dy=int frac{1}{x} dx$?

Answers to this question Homogeneous differential equation $\frac{dy}{dx} = \frac{y}{x}$ solution? assert that to find a solution to the differential equation $$\dfrac{dy}{dx} = \dfrac{y}{x}$$ we may rearrange and integrate $$\int \frac{1}{y}\ dy=\int \frac{1}{x}\ dx.$$ If we perform the integration we get $\log y=\log x+c$ or $$y=kx$$ for constants $c,k \in \mathbb{R}$. I've seen others use methods like this before too, but I'm unsure why it works.

Question: Why is it legitimate to solve the differential equation in this way?

Answer

You start with
$$y'=\frac{y}{x}\implies \frac{y'}{y}=\frac{1}{x}\implies\int\frac{y'dx}{y}=\int \frac{dx}{x},$$
and you make the change of variables in the first integral, which results in what you've written

$$\int\frac{dy}{y}=\int \frac{dx}{x}$$