I just noticed that problem 1 on Homework 11 as stated has in fact no solution -- the right-hand side term of Poisson's equation is not consistent with the given boundary condition. I have posted an updated version of the solutions here which explains exactly what is at issue here. (I have also added a bit more explanation about the forward and inverse Fourier transforms used in the solution to problem 2.)

There would be a solution if, for example, the boundary condition were changed to a homogeneous Neumann condition, or if the right-hand side of Poisson's equation were changed to (pi^3)/8 sin pi x sin pi y sin pi z. (It would be a good exercise to work through the explanation in the solutions to see why both of these problems have solutions.)

I sincerely apologise for this! Let me know if you have any questions.

Nathan