What's exactly the deal with differentials? (Confessions of a desperate calculus student)

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.