So if is the Heaviside function defined by
then you could write your RHS function
(Technically, you need to use a strict inequality for one or the other of your function pieces for - the way I've done it assigns )
So now your DE appears as
subject to and
So now we take the LT of both sides:
Did you follow that?
I'm not sure you did your LT correctly, especially the RHS.
Does this make sense?
[EDIT] There are corrections to both the RHS and LHS below.