# Homework Help: PDE, 2D Laplace Equation, Sep. of Variables, Finding Potential

1. Feb 19, 2012

### trncell

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

2. Relevant 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.

3. 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.

Last edited by a moderator: Feb 19, 2012
2. Feb 19, 2012

### vela

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