Electric potential inside rectangular hollow conductor I have a question about  a task: 
Find the potential inside a hollow, rectangular conductor, 3 sides of the conductor are grounded and one's held at a constant potential $U_0$ . 

I am looking for solutions of $\nabla^2U=0$, with the method of separation of variables. With the following  boundary conditions :     $U(x,0)=0$, $U(b,y)=0$, $ U(-b,y)=0 $ , $U(x,a)=U_0$  .
The general solution is then: $U(x,y)=(A_0+B_0x)(C_0+D_0y) + \sum_{k\neq 0} (A_k e^{kx}+ B_k e^{-kx}) (C_k \cos(ky) + D_k\sin(ky))$
And then the solution in the book says : " Let's look at the first boundary conditon : 
$ 0 = (A_0+B_0x)C_0 + \sum_{k\neq 0} (A_k e^{kx}+ B_k e^{-kx}) C_k$
if this should hold for all x, then $C_0=C_k=0$ ." My question is: Why can't I somehow find the appropriate constants so that $\sum_{k\neq 0} (A_k e^{kx}+ B_k e^{-kx}) C_k =-(A_0+B_0x)C_0$ and therefore wouldn't have to require that  $C_0=C_k=0$ ?
 A: Because $C_0=0$ and $C_k = 0$ are the appropriate constants. You can think of it like this: when you impose the boundary on the $y= 0$ face of the wire you will end up with a expression that looks like this
$$
C_0 f(x) + \sum_k C_k g_k(x) = 0 \quad -b \le x \le b
$$
That is, for every $x$ in this equation must hold, since $f$ and $g_k$ are linearly independent, the only solution to this equation is when the coefficients are identically 0
A: I find it's easier to start boundary problems like this with a sketch of the field lines and equipotential lines.  
The rules of this sketch are simple, sketch field lines to cross equipotential lines at right angles, and sketch equipotential lines cross field lines at right angles.  (I know, the two rules imply each other.) 
Normally, the equipotential lines don't cross, but the upper left and upper right corners have indeterminant values.  In this case, all out equipotential lines are going to start and end at those corners.
In the upper left and upper right corners, the field lines are going to be almost circles.  In the center, the field lines are going to be close to what we see with a standard parallel plate capacitor.


The sketch comes out to something like this.  
y is definitely a function of x, so this is why separation of variable in terms of X(x) and Y(y) isn't going to work out.
That might be a mistake.  I'll have to look at separation more closely.  Something is amiss, though.  I think it might be the indeterminate values at the upper corners. 
