For case 1, this changes our X(x) equation a little bit.

We regularly solve by seperating and finding lambda using

And we use the boundary conditions to compute. Now, we have a new set of boundary conditions we must use

and

So,

Noting that -a is a constant, we would normally remove this as it will be included in our summation at the end. But in this case it serves a crucial purpose.

This is true when

and

Of course then we have

We can then find what a makes this equation work. Of course we then continue from here, noting that in addition to the regular summation we must add

to satisfy the initial conditions.

I will explain how to do the second condition tomorrow when im not so tired. Also, I would like to add I've never delt with this boundary condition. I have delt with

and

Which makes it a lot easiar to understand what's going on. In this particular case I kept in the a instead of discarding it. I don't know if this is absolutely correct or not...Anyways, it's late bed is calling.

Cheers