Any help will be appreciated!

I am solving Laplace's Equation in 3-Dimensions using superposition.

The problem is that I don't know how to use orthogonality to solve for the constant when you have two Eigen functions.

$\displaystyle \frac{\partial ^{2}u}{\partial x^{2}}+\frac{\partial ^{2}u}{\partial y^{2}}+\frac{\partial ^{2}u}{\partial z^{2}}=0$

$\displaystyle u(x,y,z)=v(x,y,z)+w(x,y,z)$

This is my attempt to get a solution for v(x,y,z).

$\displaystyle v(x,y,z)=A(x)B(y)C(z)$

The solution using the given homogeneous boundary conditions comes out to be:

$\displaystyle v(x,y,z)=\sum Ksinh(\lambda z)sin(n\pi x)sin(m\pi y)$

where $\displaystyle n:1\rightarrow \infty$ and $\displaystyle m:1\rightarrow \infty$

The remaining boundary condition is:

$\displaystyle v(x,y,1)=1$

Which gives:

$\displaystyle v(x,y,1)=\sum Ksinh(\lambda)sin(n\pi x)sin(m\pi y)=1$

How do I use orthogonality to find K when I have two eigen functions in the solution?