5. 1) Find solution of the Laplace Equation

in the domain subject to boundary conditions:

and

2) Sketch the graph of solution for and

a)

b)

Solution:

Laplace’s Equation. Basic case: 2 b.c. are homogeneous, and only one b.c. in y.

1) Separation of variables:

Sturm-Liouville Problem (equation with two homogeneous b.c.):

Second equation:

Use exponential form of solution:

Solution has to be bounded

Therefore

Write the solution in the form of the infinite series:

Apply the last b.c. at y=0 to determine coefficients :

Treat it as the cosine Fourier series expansion of function , then coefficients are:

Then solution is:

3) Case

then

for (orthogonality)

And solution becomes: