Your original conclusion that this is a seperable equation and that you must use partial fractions on the R.H.S is correct.
Let's just work through it.
So the R.H.S becomes
So far so good.
Now the L.H.S
L.H.S = R.H.S
Clearly to bring down Y we would use ln as you suggested. Is there any problem with having (as an example)
Well, let us think about this. What is
Well, this will evaluate to a number! So essentially, we'll have ln(some number) which is clearly no problem. So to answer your question, this is not a problem.
But you should note for
We have the restriction that,
Other then that, we are fine!