Handling $\frac{dy}{dx}$ as a ratio.

Question

I know that given a differential equation, one that is separable, it is not fully correct to handle the $\frac{dy}{dx}$ as a ratio. Meaning that it is not simply a small difference in $y$ over a small difference in $x$, as the difference is infinitesimal. It is simply a notation created by Leibnitz that is equivalent to $f'(x)$.

My question is if this is the case then why is this method for ODE still used and taught? Is the answer always correct when dealing with ODE using this method or is does it lead to wrong results in some cases?

Answer 1

When a seemingly illegal mathematical manipulation has been assigned a logically consistent foundation and when the users are fully aware of what they are doing, then the manipulation is adopted for convenience. There is a handy example: the symbol"$dx$" in a definite integral $\int_{a}^{b}f(x)dx$. One can argue that writing $\int_{a}^{b}f$ suffices to preserve all the things relevant here; so why the "redundant" one is still in use? A reason is that it is mnemonic and hence convenient! It reminds people of the construction of (Riemann) integration. Once the foundation of limit is solid, mathematicians won't be confused by the mysterious "dx" anymore; instead, they use symbols wisely not pedantically. The seemingly abuse of symbol sometimes does in fact help, as long as one knows his stuff.

So don't be played by symbols; play the symbols.

Answer 2

One can define a notion of differential: $\mathrm{d}x$ and $\mathrm{d}y$ are meaningful objects on their own, and when $y$ can be expressed as a differentiable function of $x$, we really can talk about their ratio, in the sense that $\frac{\mathrm{d}y}{\mathrm{d}x}$ means the unique function that fits in the formula

$$ \mathrm{d}y = \frac{\mathrm{d}y}{\mathrm{d}x} \mathrm{d}x$$

This notation is still taught, because doing calculus with related quantities (as opposed to calculus with functions) is both extremely useful and commonly used, and differentials are by far the best notation for expressing differential calculus in terms of related quantities.

Thus, even though introductory calculus classes tend not to talk about related quantities (instead showing how to translate such things in terms of functions), they still need to demonstrate how to use the notation so that students are prepared when they encounter it. Or even so that the book can use the notation.