1. Does the function  satisfy Laplace’s equation?

Yes, a direct calculation shows

2.Find an expression for the potential in the region between two infinite parallel planes  on which the potentials are respectively given by  and 0.

In this case also we have solutions which are written as products of separate variables. The solution can be written as (since  X and Y are hyperbolic functions, the function Z is a linear combination of sine and cosine),

The constants A and B can be determined by the boundary conditions. For  z=0,  gives B=1.  For  gives A=0. Thus

 

3. Find an expression for the potential in the region between two infinite parallel planes, the potential on which are given by the following :

The z dependence of the potential is a linear combination of hyperbolic sine and cosine function with argument . Thus the general expression is

For z=0, the  potential being  zero, the cosh term vanishes, A=0.    gives  so that