Laplace in rectangular coordinates, u(x,y)=X(x)Y(y):
which has the solution:
note:These are good for a semi-infinite plate that has oscillating solutions in
x and tends to zero as y goes to infinity. Could equally well have a problem
with the opposite equations in x and y (i.e., here you would change the
sign choice of 
.
Note on the Diffusion Equation u=F(x,y,z)T(t):
which for a one dimensional problem has solutions:
Laplace in cylindrical coordinates, 
:
which has solutions:
Laplace in spherical coordinates, 
:
which has the basic solutions:
Remember that your solution has to show the symmetry of the problem!
Also remember that you want your solutions to be physical,
i.e. non-divergent for the geometry observed.
Keep in mind
that any orthogonal function 
is even for even n and odd for
odd n!