The following question is asked without malicious intentions - it's not intended as a flamebait!

In my physics textbooks (Young & Freedman in particular) I have often seen derivations of equations that use multiplication of, say, $dx$ on both sides, and then integrating.

Taking courses both at the math and physics department I've noticed that real mathematicians often frown at this. Why is that?

Suppose you have the equation
$$f(x) = g(x).$$
You could "multiply both sides by $dx$"
$$f(x) \, dx = g(x) \, dx$$
and then integrate over some interval
$$\int_a^b f(x) \, dx = \int_a^b g(x) \, dx.$$
However, unless you have learned about differential forms, the second equation above is meaningless. This can easily be avoided by simply integrating both sides of $f(x) = g(x)$ with respect to $x$ just as you learned in calculus.

Now consider the differential equation
$$f(y) \frac{dy}{dx} = g(x).$$
Here, many people will "multiply by $dx$" to get
$$f(y) \, dy = g(x) \, dx$$
and then integrate both sides. Again, without the machinery of differential forms, this statement is meaningless. Here we can avoid this by integrating the original equation with respect to $x$:
$$\int f(y) \frac{dy}{dx} \, dx = \int g(x) \, dx$$
or to use different notation,
$$\int f(y(x))y'(x)\, dx = \int g(x) \, dx.$$
Then as long as certain hypotheses are satisfied, the change-of-variables (substitution) theorem says that this is just
$$\int f(y) \, dy = \int g(x) \, dx$$
so we are back to where the first method lead us.

The point of this is that manipulations involving "multiplying by $dx$" are usually shorthand for some more "rigorous" method. However, if you are aware of (and comfortable with) the mathematics going on behind the scenes, then there is no loss of rigor. "Real mathematicians" don't frown upon using these kinds of manipulations themselves, they just frown upon students using them without knowing why they work.
