TITLE: Separating the variables possible?
QUESTION [1 upvotes]: Given the differential equation $$ \frac{dy}{dx} = y+x$$
I am told this differential equation is separable. Meaning I need to rewrite the RHS into a product of two variables depending on y and x.
I've tried for some time now but I simply cannot figure out how this is separable. I'm able to solve it using the method of "integrating factor" and so I know the solution should be $$ y = Ce^x-x-1 $$
Any ideas?

REPLY [3 votes]: Perhaps with $z:=y+x+1$,
$$ \frac{\mathrm dz}{\mathrm dx}= \frac{\mathrm dy}{\mathrm dx}+1=y+x+1=z.$$
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