I am trying to solve the following:
I can separate this into two different problems: $\theta(x,y)=v(x,y)+w(x,y)$.
Let's call one of them A:
While the other can be B:
On the other hand, Professor's first step in the solution is to say that $\theta(x,y)=2x-1+w(x,y)$. So I think he got the solution for v(x,y) substituted into the equation. I have tried to get there myself but...
Going for A, by separation of variables:
so the equation turns into $-XY''-X''Y=0$
So I think I have to solve for Y first:
And solving the eigenvalue problem I get:
$\lambda_n=(n\pi)^2; Y_n(y)=\cos(n\pi y)$
Then, I continue trying to solve for X(x):
I can substitute $\lambda$ so $X''=(n\pi)^2 X$. I have noticed that $\lambda_0 = 0 $ (n=0) is an eigenvalue and I get the eigenfunction $X_0(x)=2x-1$ which is OK with the solution, but I don't know how to prove that that is the only one.
I would really appreciate a piece of advice. Thank you very much!