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In the notation for integrals, I was taught that we use a "dx" at the end as a delimiter and to tell us what variable we are integrating with respect to, and that it is not a quantity being multiplied by the integrand. But in differential equations, we multiply the denominator of dy/dx and receive dx multiplied by some function on the right hand side (what does that even mean?) and then we integrate that expression, which seems weird to me if dx isn't a factor of the integrand. What's going on here?

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  • $\begingroup$ This question might be helpful: math.stackexchange.com/questions/47092/… $\endgroup$
    – teddy
    Commented May 5, 2020 at 0:29
  • $\begingroup$ And this and this $\endgroup$ Commented May 5, 2020 at 0:48
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    $\begingroup$ In undergraduate differential equations courses, calculations that involve manipulating $dx$ and $dy$ as independent quantities can always be rephrased easily to avoid doing so. (That said, "infinitesimal intuition" should not be thrown out the window. For intuition, we can think of $dx$ and $dy$ as tiny but finite quantities and do calculations with them, obtaining approximate equalities which "in the limit" become exact equalities. Infinitesimal intuition is very clear and powerful, so it's not hard to see why physicists and other people -- including mathematicians -- might use it often.) $\endgroup$
    – littleO
    Commented May 5, 2020 at 0:52
  • $\begingroup$ @teddy The question mentions differential forms - is that what dx and dy are "formally" in the context of my question too? $\endgroup$
    – hawexp
    Commented May 5, 2020 at 0:59
  • $\begingroup$ @hawexp I'm not sure since I don't have the mathematical background/formalism in that field, but that answer also avoided using differential forms, which I thought might be helpful. $\endgroup$
    – teddy
    Commented May 5, 2020 at 2:58

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If I understand correctly, you mean if we have something like: $$\frac{dy}{dx}=f(y)g(x)$$ then we get: $$\int\frac{1}{f(y)}\frac{dy}{dx}dx=\int g(x)dx$$ writing it like this shows that we integrate wrt the same variable on both sides but it can be simplified to: $$\int\frac{dy}{f(y)}=\int g(x)dx$$ similarly if we have an expression of the form: $$f(x,y)dx+g(x,y)dy=0$$ we can view this as: $$\lim_{(\delta x,\delta y)\to(0,0)}f(x,y)\delta x+g(x,y)\delta y\to0$$ whilst this is not exactly true since the variables are dependent I think it helps to understand what they represent

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