Note: Use the prime notation for derivatives, so the derivative of is written as . Do NOT use . The variable is typed lambda.

Solve Laplace's equation with homogeneous vertical boundary conditions:

We try for a separable solution , plugging into the PDE for we get

The PDE can be separated into an ODE in X and an ODE in Y (put all multiplicative constants in the DE for Y):

= =

where is a constant. The differential equations can be separated because they are independent of each other.

The boundary conditions in the PDE translate into initial conditions for the differential equation in X:

DE in X:

Initial Conditions (IC's) in X:

We have solved this S-L problem before. For each there is, up to a multiplicative constant, one basic solution, , which is a trigonometric function with period :


The corresponding to this solution is . With , the corresponding differential equation for is:

DE in Y:

Using A and B as the constants, the solution for Y with is a linear combination of hyperbolic functions


The PDE is linear so it admits a series solution that is a linear combination of the solutions

Now we can apply the lateral boundary conditions:

The first series we recognize as a Fourier series on [0,a], so



The second series is also a Fourier series with coefficient



Which we solve for


You can earn partial credit on this problem.