Solve Laplace's equation with homogeneous vertical boundary conditions:

We try for a separable solution

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

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

The

DE in Y:

Using A and B as the constants, the solution for Y with

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

---------- | dx |

The second series is also a Fourier series with coefficient

---------- | dx |

Which we solve for

----------------------------------- | ---------- |

You can earn partial credit on this problem.