[bikenaga]

> t acts like a mnemonic scribble for an intermediate step. It doesn't have any mathematical meaning.

That's about right. When you cover elementary solution methods for differential equations, you start with something like y = dy/dx. You're supposed to separate variables ("get the x's on one side and the y's on the other, then integrate"). So it's tempting to just write "dy/y = dx", even though as you say it doesn't have any mathematical meaning. But it's helpful in keeping track of the algebra. You then forget you wrote that meaningless but helpful step and write "∫ dy/y = ∫ dx" which is okay, and go from there.

Looking at one of my old diff eq books I see whole sections where this kind of casual algebra with differentials is the norm.

When anyone would ask me about this in class, I'd say something like this. Think of a solution curve for y = dy/dx as a parametrized curve, so x = f(t) and y = g(t). Then interpret the equation as y = (dy/dt)/(dx/dt), write dx/dt = (1/y) (dy/dt), then integrate both sides with respect to t:

  ∫ 1/y (dy/dt) dt = ∫ (dx/dt) dt

Change variables to get "∫ dy/y = ∫ dx". After a while we believe that this will always work, and we just suppress the stuff about parametric equations.