When we start learning about differential equations sometimes we "multiply" both sides of the equation by a differential and then integrate.
Example: $\frac{dy}{dx}=x$ then $dy=x*dx$ and so on.
I have always thought this is a "shortcut" since multiplying by a differential doesn't make a lot of sense to me. But, why does it work? What are we really doing?
EDIT:
I'll clarify a little more. My specific question is: Why can we treat differentials as real numbers?
$x$ and $y$ can be thought of as each being functions of some independent parameter $t$. The chain rule tells us that, $$ \frac{dy}{dx}=\frac{dy/dt}{dx/dt}$$,
So when we see
$$ dy/dx = x $$
we can write
$$ dy/dt = x dx/dt$$
then of course we can integrate both of these expressions with respect to $t$
$$ \int \frac{dy}{dt} dt = \int x \frac{dx}{dt} dt$$
then the integration can be accomplished by a change of variable.
So when we write those lone differentials we are really just ignoring the parameter because it is completely redundant for the end results. There are some difficulties that must be avoided, for instance $y(t)$ and $x(t)$ should always be monotonic, but these details don't change the main idea.
