After some calculations in a PDE I have the following equations:

For the first equation, there are 3 possible cases:

(i) is positive

(ii) is negative

(iii) equals 0.

I also have the boundary conditions

I was hoping someone could answer some general questions I have about these cases.

First of all, by starting with case (iii) we have:

Then with BC's I get

so is a solution.

Also,

so ......my question here is if leads to a solution as just shown, does this automatically mean that need to be included in my fourier series solution to the PDE?

Now for case (i):

leads to solutions

so with BC's

Now leads to a trivial solution, so

for

and

so

For the third case,

Now I get

Apparently this should be a trivial solution, but I don't understand how it is different from case (ii). I would get

so

What am I missing here? How can I have two values for ?

Thanks to anyone who can help me understand this.