Seperation of variables justification?

Question

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?

Since you mention dividing by $f(x)$ only, I'm assuming you meant for $g$ to be a function of $t$, hence the edit. If you meant otherwise, let me know or feel free to roll-back the edit. — amWhy, Feb 12 '14 at 13:19
Oops, yes that's what I meant. Since you've pointed it out though I'll edit to use the usual notations. — GPerez, Feb 12 '14 at 13:23
Up to something you should justify by a uniqueness theorem, things are even easier: $\frac{1}{f(y)}\frac{dy}{dx}$ is seen as the derivative of $G \circ y$, where $G$ is a primitive of $1/f$. That's all. — Siminore, Feb 12 '14 at 13:30
possible duplicate of What am I doing when I separate the variables of a differential equation? — Orat, Feb 12 '14 at 13:34

score 1 · Answer 1 · answered Feb 12 '14 at 13:48

1

The function $y=y(x)$ has a differential as $$dy=\frac{dy(x)}{dx}dx \\ dy=f(x)g(y)dx$$ One more thing you could easily think of $dx$ as a small quantity in your courses not as a $1$-form as others do in differential forms.

answered Feb 12 '14 at 13:48

Semsem

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Seperation of variables justification?

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