thermo-ish diffusion/wave equation - metal plate and temperature difference

samee

1. The problem statement, all variables and given/known data

The edges of a thin plate are held at the temperature described below. Determine the steady-state temperature distribution in the plate. Assume the large flat surfaces to be insulated.

If the plate is lying along the x-y plane, then one corner would be at the origin. The height of the plate would be 1m along the y-axis and the length would be 2m along the x-axis. The edge along the y-axis is being held at 0 C. The edge along the x-axis is being held at 0 C. The edge parallel to the x-axis is being held at 0 C. The edge parallel to the y-axis is being held at 50sin(pi*y) C.

2. Relevant equations

So I'm assuming this question is actually just a diffusion equation or a wave equation, because that's what the rest of our homework was on. Alpha²u_xx=u_t
and
u(x,t)=X(x)T(t)=(C₁coskx+C₂sinkx)e^-K2alpha2t+C₃+C₄x

3. The attempt at a solution

So I tried to solve this like the wave equations and it seems to just be blowing out of proportion and not making sense... Also... I think we need to consider a thrid position variable here, we need x,y AND t. I don';t know how to do this at all :(