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I'm reading on my analysis book that "the first order differential equation $y'=f(x,y)$ can be rewritten as $dy-f(x,y)dx = 0$ and that it is equivalent to the more general equation $L(x,y)dx+M(x,y)dy = 0$". Really I don't understand how can be made this "transformation" without thinking the derivative as a ratio of infinitesimal. Thanks in advance

C.Baroni
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  • Use parametrization: $$x=x(t) \ y=y(t)$$ Then consider a function $u(x(t),y(t))$ and differentiate w.r.t. $t$ – Yuriy S Oct 25 '18 at 22:55
  • This is confusing in part because of notation. What can we take derivatives of? Functions is the answer. So if we have a function $f$, we can differentiate and obtain $\frac{d}{dx}f$. So if we agree that you can only take derivatives of functions, what does it mean to write $y; '$ or $\frac{d}{dx}y$? Is $y$ a function? It isn't. Given a function function $f$, we often write $y = f(x)$, which is an equation satisfied by some set of points ${(x,y)}$. $y$ in general is a coordinate, but only a certain coordinates satisfy $y = f(x)$. We get away with it because $\frac{d}{dx}f$ is precisely – DWade64 Oct 25 '18 at 23:02
  • How the output varies with the input (look at the limit definition). And the equation $y = f(x)$ is basically saying (or equivalently saying) that you are calling the output of $f$ the letter $y$. With this in mind, we can say $\frac{d}{dx}f$ (which is how the output varies with the input) as $\frac{d}{dx}y$. Now, differentials have the same problem as derivatives. You can only apply the differential $d$ to functions $f$, just as you can only apply the derivative $\frac{d}{dx}$ to function $f$. Define $df := \frac{d}{dx}f \Delta x$ – DWade64 Oct 25 '18 at 23:06
  • Given two functions $f,g$ and their plots $y = f(x)$ and $y = g(x)$, now we definitely can't write $\frac{d}{dx}y$ because it's ambiguous. See here for more relating to my comment and the answer already given which is a nice answer (note that the differential of the identity function $h: x \mapsto x$ is $1\Delta x$ – DWade64 Oct 25 '18 at 23:17

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Even though it might seem like so at first, you don't need to think of differentials as infinitesimals and derivatives as their ratios to make sense of those expressions.

You can define a differential $\text{d}y$ as the linear part of a variation $\Delta y = f(x+\Delta x)-f(x)$, or as an abstract mathematical object known as a differential form. Either way, you would get that (for one-variable differentiable functions)

$$\text{d}y = f'(x) \text{d}x = \frac{\text{d}y}{\text{d}x}\text{d}x$$

For a more in-depth explanation, you can check out my answer to this question here: When can we not treat differentials as fractions? And when is it perfectly OK?; or any other answer there for that matter.

TeicDaun
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