PDE, 2D Laplace Equation, Sep. of Variables, Finding Potential

[trncell]

Homework Statement

A square rectangular pipe (sides of length a) runs parallel to the z-axis (from -\infty\rightarrow\infty). The 4 sides are maintained with boundary conditions
(i) V=0 at y=0 (bottom)
(ii) V=0 at y=a (top)
(iii) V=constant at x=a (right side)
(iv) \frac{\partial V}{\partial x}=0 at x=0 (left side).
Use separation of variables to find the potential V(x, y, z) inside the pipe.

Homework Equations

General solutions for 2D Laplace equation
X(x)=Ae^{kx}+Be^{-kx}\qquad Y(y)=C\cos ky+D\sin ky where A, B, C, and D are constants.

The Attempt at a Solution

I know since that C=0 in order to satisfy the boundary conditions V=0 for y=0 and y=a. I however do not know how to satisfy the boundary condition for the left side where dV/dx=0.
This is what I've tried:
\frac{d}{dx}[Ae^{kx}+Be^{-kx}]=kAe^{kx}-kBe^{-kx}
and set A=0 because kAe^{kx} does not go to zero.

 

[vela]

Set the derivative equal to 0. This gives you a relationship between A and B.