I'm teaching ordinary differential equations for the first time, and I would like to give a compelling visual explanation of why it makes sense to "multiply by $dx$" and integrate when you want to solve a separable equation like $\frac{dy}{dx} g(y) = f(x)$.
Roughly what came to mind: one can use the tangent slope of a solution to the equation to give two congruent right triangles with "adjacent" and "opposite" side lengths $(\Delta x, \Delta y)$ (approximately) and $(g(y), f(x))$. Using the fact that these are similar triangles, we see that cross multiplying yields $g(y)\,\Delta y = f(x) \,\Delta x$ (really approximately), and these are approximations of the areas represented by $\int g(y) \,dy$ and $\int g(x) \,dx$.
Question: Are there other visual explanations for the method of separation of variables? Perhaps one that can make a more direct connection with areas and/or are more precise?
I urge you not teach your students to multiply with $dx$ as this can cause conceptual problems for weaker students in the future. Specifically, $dx$ is not a real number, but represents part of a limiting process. I recommend that you stress the application of the chain rule of differentiation. Specifically given \begin{equation} f(x) = g(y(x)) y'(x) \end{equation} I would instead stress the need to find a function $G$ such that \begin{equation} \frac{dG}{dy}(y) = g(y) \end{equation} so that the chain rule could be applied, collapsing the equation to \begin{equation} f(x) = \frac{d}{dx} \left[ G(y(x)) \right] = g(y(x))y'(x) \end{equation} which will motivate your students to find an integral for $f$.