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:

or

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.