Either with or with will reduce a first-order homogeneous equation to a separable one. Here's a quote from Zill, 6th Ed., p. 55:
Note that in Zill's notation, the DEAlthough either of the indicated substitutions can be used for every homogeneous differential equation, in practice we try whenever the function is simpler than .
is the homogeneous DE under discussion.
Does that answer your question?