In differential equations, the idea of multiplying by an infinitesimal, dx, is used in the method of separable equations. My confusion is in why this works as I've heard many times in the past that you shouldn't multiply and cancel out infinitesimals like that. I understand that in the setting of nonstandard analysis that there may not be something wrong with this, but in the usual setting is there a more rigorous interpretation of this?
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1Hint: chain rule. – Simply Beautiful Art Dec 15 '19 at 18:54
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1I was going to write an answer, but then searched and saw that the accepted answer to this question directly explains what the OP is looking for. – Axion004 Dec 16 '19 at 02:45
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Multiplying by $dx$ is not the only way to interpret what's happening when you use separation of variables. Consider:
$$ \frac{dy}{dx} y = x $$
It's commonly said that the next step is to 'multiply by $dx$ on both sides' to get $y \ dy = x \ dx$ and then integrate. It's much more understandable at the basic level to say: Integrate both sides with respect to $dx$. Then you have:
$$ \int{y \frac{dy}{dx} \ dx} = \int{x \ dx} $$
Which simplifies to
$$ \int{y \ dy} = \int{x \ dx} $$
Now there's no issues with interpreting the differential; you can use the fundamental theorem of calculus instead.
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