# If dy/dx is an operator, what permits us to seperate it when solving separable differential equations?

I mean, intuitively I see it has something to do with gradients in general, but with all the "with respect to" and "change in", my head gets a little dizzy.

Could someone please spell it out a bit clearer for me? Why can we all of a sudden (for instance) multiply both sides by dx???

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• Awms A
Lv 7

First, just a technicality, but dy/dx isn't an operator... d/dx is.

Your question still stands though... why can we "multiply both sides by dx" when the dx is just notation in the operator? Well, it's just for convenience... note that we could do an entire problem without ever making this step...

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Consider

dy/dx * xy = 1

First divide both sides by x:

dy/dx * y = 1/x

Now note that the left hand side can be rewritten as

d/dx ( 1/2 y^2 ) = 1/x

Now integrate both sides with respect to x to get

1/2 y^2 = ln|x| + C

and this is the same thing we'd get if you had "multiplied both sides by dx".

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It may not make sense still, but that's really it. The notation is made so that the chain rule and integration are compatible with one another.

What we're actually doing is integrating with respect to x, and then using substitution to integrate the side with the y-variables.