I would guess so. I haven't really "gone over" your other math leading up to the expressions with A and B, though.
Btw, I think that my guess about using images assumes that both situations have a unique solution. That is not always the case for all PDEs. I've seen some cases where a unique solution is only obtained after imposing e.g. a restriction on the growth of the solution at large x. If solutions are not unique in your case, you might need to impose such a condition, e.g. large-x behaviour would then imply A=0.
This is like saying that you are searching for solutions within some more restricted function space, which could be bounded functions, L^2 functions, etc.
I don't really remember much about the theorems on uniqueness of solutions for the different PDEs...
Anyway, good luck!