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This question already has an answer here:

Even if many mathematicians don't like the notation, I have found in many rigorous math books things like

$\frac{dy}{dx}=Ay$

so

$\frac{dy}{y}=Adx$

What I don't understand is this: under which conditions is ok to treat infinitesimal as numbers ? (multiplying for example both sides of an equation by dx).

Edit:

Not only, what allows me to integrate both sides of the last equation ???

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marked as duplicate by Mark S., Hans Lundmark, Kemono Chen, Gibbs, Dylan Feb 11 at 14:34

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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If you want to make this rigorous you can introduce a number system similar to that of dual numbers, viz. $dx^2=0$. (See also here.) Depending on the calculus task at hand in more advanced examples, you might change these axioms slightly. For example, a metric $ds^2=g_{\mu\nu}dx^\mu dx^\nu$ would use $dx^\mu dx^\nu dx^\rho=0$, while Brownian noise in stochastic calculus may be taken to satisfy $dW_t^2=dt,\,dW_t^3=0$.

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