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I can't understand why we multiply by $dx$ in differential equations to integrate? People say so that we know what are we integrating with respect to, but this is obviously not true right?

What does $\;x\,dx = (y-y^2)\,dy\;$, for example, mean?

If $dx$ or $dy$ are very very small with a limit that goes to zero how can we multiply by zero?

What's a differential?

This is not an exact duplicate. I still don't know what a differential is.

Mohamed Ayman
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  • The easy version is: Writing $x dx = (y-y^2)dy$ is just a short (symbolic) way of writing the two differential equations: $y'(x)=\frac{\partial y(x)}{\partial x}=\frac{x}{y(x)-y(x)^2}$ and $x'(y)=\frac{\partial x(y)}{\partial y}=\frac{y-y^2}{x(y)}$. In that way this is just a slightly confusing notation, that I don't really like. However the hard version is: it can be made precise by considering differential forms. https://en.wikipedia.org/wiki/Differential_form – mjb Jun 30 '13 at 20:33
  • An example of this is: the circle (defined by $x^2+y^2=1$) does satisfy the equation $y'(x)=\frac{-x}{y(x)}$ where $y \neq 0$ and $x(y)=\frac{-y}{x(y)}$ where $x \neq 0$. The notation for that is $xdx+ydy=0$. – mjb Jun 30 '13 at 20:38

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