Solving for the steady state potential distribution for a square with boundary conditions

Question

I am struggling on solving for the steady state distribution of potential for my system.

It is for a square geometry. On the left vertical side $(x = 0)$ , the potential is $-10$. On the right vertical side $(x = a)$, the potential is $10$. The top and bottom of the square are both $20x - 10$ equals the potential $(y = a$ and $y = 0$ , respectively $)$.

So, mathematically the initial values are:
$V(x=0,y) = -10$
$V(x=a,y)=10$
$V(x,y=0)=V(x,y=a)=20x-10$

I am using this formula to solve for the coefficients given the boundary conditions:

$$V(x, y) = [A\sin(kx) + B\cos(kx)][Ce^{ky} + De^{-ky}] $$

The part I am struggling with is solving for the coefficients. I am thinking it might be quite complicated. I would appreciate any help, thanks.

P.S.

What I have done so far is break it up into different parts. For example, I solved for the coefficients for the case where for $y = a$ and $y = 0$ the potential is zero, then for $x = 0$ and $x = a$ the potential is negative ten and ten, respectively. Then, I was going to consider, i.e. solve for the coefficients, for the case $y = a$ and $y = 0$ the potential is $20x - 10$ . Then I was going to superimpose the two parts and get an overall solution. Thanks!

Answer 1

I am not sure I understand the question. You don't tell us anything about the conductivity inside the square, and the potential can depend on it. Or, maybe you could just tell us for what equation you are trying to get a solution? For some value of uniform conductivity you can just have the potential 20x-10 for each point. Or maybe you have in mind something completely different - the tag "electrostatics" seems strange.