What's exactly the deal with differentials? (Confessions of a desperate calculus student) So I don't know if I'm the only one to feel this, but ever since I was introduced to Calculus, I've had a slight (if not to say major) aversion to differentials. 
This sort of "phobia" started from the very first moment I delved into integrals. Riemann sums seemed to make sense, though for me they were not enough for justifying the use of "dx" after the integral sign and the function. After all, you could still do without it in practice (what's the need for writing down the base of these rectangles over and over?). I was satisfied by thinking it was something merely symbolic to remind students what they were doing when they calculated definite integrals, and/or to help them remember with respect to what variable they were integrating (kinda like the reason why we sometimes use dy/dx to write a derivative). Or so I thought.
Having now been approached to differential equations, I'm starting to realize I was completely wrong! I find "dy" and "dx" spread out around equations! How could that be possible if they are just a fancy way of transcribing derivatives and integrals? I imagined they had no meaning outside of those particular contexts (i.e.: dy/dx, and to indicate an integration with respect to x or whatever).
Could anybody help me out? I'm really confused at the moment. I'd really appreciate it :)  (P.S.: Sorry to bother you all on Thanksgiving - assuming some of you might be from the US.)
EDIT: I don't think my question is a duplicate of Is $\frac{\textrm{d}y}{\textrm{d}x}$ not a ratio?, as that one doesn't address its use in integrals and in differential equations. Regardless of whether dy/dx is a ratio or not; what I'm really asking is why we use dx and dy separately for integration and diff. equations. Even if they're numbers, if they tend to 0, then dx (or dy) * whatever = 0. Am I wrong in thinking that way?
 A: If you read a real analysis textbook such as Calculus by Spivak, they manage to develop calculus rigorously while avoiding differentials like "$dx$" and "$dy$" entirely.  This is the standard way to make calculus rigorous -- you just avoid using differentials.  And indeed, in undergrad differential equations classes, arguments that involve manipulating $dx$ and $dy$ as individual quantities can easily be rephrased to avoid doing this.
For example, if a differential equations textbook says:
\begin{align}
& y \, dy = dx \\
\implies & \int y \, dy = \int \, dx \\
\implies &  \frac{y^2}{2} = x + C \\
\end{align}
we can rephrase this argument as
\begin{align}
& y \frac{dy}{dx} = 1 \\
\implies & \frac{y^2}{2} = x + C,
\end{align}
where in the second step we simply took antiderivatives of both sides, using the chain rule in reverse to find an antiderivative of $y \frac{dy}{dx}$.
But note: even though a rigourous approach might avoid using differentials entirely, there is no need to throw "differential intuition" out the window, because it makes perfect sense if we just think of $dx$ and $dy$ as being extremely tiny but finite numbers, and if we replace $=$ with $\approx$ in the equations we derive.  Perhaps the word "infinitesimal" could be thought of as meaning "so tiny that the errors in our approximations are utterly negligible".  We can plausibly obtain exact equations "in the limit" (if we are careful).
There is something aesthetically appealing about treating $dx$ and $dy$ symmetrically, which can perhaps in some situations give us a feeling that the approach using differentials is the "right" way or more beautiful way to do these computations.  Compare these two ways of writing an "exact" differential equation:
\begin{equation}
I(x,y) \,dx + J(x,y)\, dy = 0
\end{equation}
vs.
\begin{equation}
I(x,y) + J(x,y) \frac{dy}{dx} = 0.
\end{equation}
The first version is aesthetically compelling, because it's more symmetrical; this might help explain why the second version is not seen more often (despite its being easier to understand, in my opinion).
Of course, for any results derived using "differential intuition", we must later find a rigorous proof to confirm there is no mistake.
Note also: There are other approaches to making calculus rigorous (based on nonstandard analysis I think) that actually make infinitesimals rigorous.  So they manage to embrace $dx$ and $dy$ as legitimate quantities, rather than avoiding them. 
Additionally, in differential geometry, quantities like $dx$ are defined precisely as "differential forms", and some treatments of calculus (like Hubbard & Hubbard) embrace differential forms at an early stage.  But you can understand calculus rigorously without using differential forms.
A: If you want to know what a mathematical thing is, you need two things:


*

*How it works - what the rules of using it are.

*A way of expressing it in terms of mathematical things you know and
trust.


For differentials, we know what the rules of using them are. And I'm told some clever folks have worked out how to model them (notably Abraham Robinson with non-standard analysis). 
There is a whole pyramid of mathematical things that we accept without demur these days that were once as suspect as differentials probably still are. What tends to happen is that we know how the thing works, then we have to find what it is or might be. 
If you're used to counting, what's a negative number? ... a rational number?
Our confidence in real numbers is perhaps as naive as that in differentials. 
