calculus - Seperation of variables justification?

I haven't found a similar question on Math SE, but I may not have looked enough because I find it hard to believe someone hasn't already asked this. Anyways, here goes:

I'm studying mathematics, but one of the courses is a course on physics. So, since my university chooses not to give courses on differential equations until we have a solid knowledge of Algebra, Geometry, Analysis, Topology, etc., the physics course includes a small supplement on ODE's. To my dismay though, one of the first things we learned was that we could solve $$\frac{dy}{dx}=f(y)g(x)$$
By multiplying by $dx$ on both sides, dividing by $f(y )$ and integrating on the left with respect to $x$, and on the right with respect to $x$. I have no clue how this even makes sense as $dy/dx$ and $dx$ or $dy$ in an integral are just notations. Could someone elaborate a justification for this process? As a side note, is there any way to discuss these things intrinsically? Or is it like calculus where we always talk about $f(x)$ and use the canonical basis?