Why does the method of separable equations work in differential equations? 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?

dredyue 2022-08-12 Answered
Why does the method of separable equations work in differential equations?
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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Answers (1)

sekanta2b
Answered 2022-08-13 Author has 17 answers
Multiplying by dx is not the only way to interpret what's happening when you use separation of variables. Consider:
d y d x y = x
It's commonly said that the next step is to 'multiply by d x on both sides' to get y   d y = x   d x and then integrate. It's much more understandable at the basic level to say: Integrate both sides with respect to d x. Then you have:
y d y d x   d x = x   d x
Which simplifies to
y   d y = x   d x
Now there's no issues with interpreting the differential; you can use the fundamental theorem of calculus instead.

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