How would I solve the following pde? I have tried using separation of variables, however I can not get to a solution and I have tried to find a solution online however there exists no resources. So I would like to understand how I would go about in solving this?

 A: The best way to solve this is to tackle each non-homogenous part separately. Define $5$ functions that solve the following equations:


*

*$u_1: \ \Delta u_1 = -x,$ all BCs $ = 0$

*$u_2: \ \Delta u_2 = 0,$ ${u_2}_y(x,0) = 1$, all other BCs $=0$

*$u_3: \ \Delta u_3 = 0,$ ${u_3}_x(0,y) = 1$, all other BCs $=0$

*$u_4: \ \Delta u_4 = 0,$ ${u_4}(a,y) = y$, all other BCs $=0$

*$u_5: \ \Delta u_5 = 0,$ ${u_5}(x,b) = f(x)$, all other BCs $=0$
All of these solutions can be found relatively easily with separation of variables with the exception of the first, for which you should probably use eigenfunction expansion method. You should be able to modify most of the analysis from here. The eigenfunctions can be found via separation of variables and they will look very similar to the functions you will be using in the expansions in the other subproblems. The sum of all of these functions will be the solution to your equation.
This problem is very tedious and annoying. I'm sorry if someone assigned this to you for homework. 
