E&M Problem using Maxwell's Equations

[shmiv]

Homework Statement

The problem statement is in the attached .png file. There are a few equations involved which would have taken a decent amount of time to type out.

Homework Equations

The Attempt at a Solution

I understand Maxwell's equations, and wrote out all 4 assuming there are no sources present. I then have 8 scalar equations that should allow me to solve for the remaining E&M field components, but I can't figure out how to relate them and arrive at a solution. Do I need to incorporate boundary conditions as well?

Thanks in advance for any and all help!

 

[Mindscrape]

Yes, you need to incorporate boundary conditions, the problem even says so.