Is it legal to multiply left side of equation by $\mathrm{d}y$ and right side by $\mathrm{d}x$?

Question

I have a question about calculus rules which I find difficult to find an answer to. Say I have an equation of $x$ and $y$, e.g.: $y=x^2$. Am I then allowed to multiply the left side by a small change in y and the right side by a small change in $x$: $y\mathrm{d}y=x^2\mathrm{d}x$? If the equation is true, then it should be true for an infinitesimal change in $x$ and $y$, but I am not sure about the Leibnitz notation and how we are allowed to use it in equations.

Answer 1

$dx$ and $dy$ are still real numbers, even if they are very small. So $y=x^2$ does not imply that $y\,dy = x^2\,dx$ unless $dx = dy$.

Answer 2

$\frac{\operatorname{d}y}{\operatorname{d}x}=2x$

so $\operatorname{d}y=2x\operatorname{d}x$

$y \operatorname{d}y = x^22x\operatorname{d}x$

verifying $\frac {\operatorname{d}y}{\operatorname{d}x}=\frac{2x^3}{y}=\frac{2x^3}{x^2}=2x$