Solving Poisson/Dirichlet PDE with Boundary Conditions and 2u Variable

[erok81]

Homework Statement

Show the two problems (i.e. give the boundary conditions and PDE's) that the given problem must be broken into in order to solve the PDE u_xx+u_yy=2u+f(x,y) satisfying the shown boundary conditions.

Homework Equations

See attachment.

The Attempt at a Solution

We have a midterm coming up in a couple days and of course were given a practice midterm. This problem came up. I can solve these if I have u_xx=f(x) or probably even u_xx+u_yy=f(x,y). I have never seen a problem that also contains a 2u. I checked my notes and my text but cannot find anything regarding how to solve with a u in the PDE.

From the image I am not sure why it gets broken up that way either. If it was just f(x) alone I'd set one problem with all zero boundary conditions equal to f(x) and the non-zero boundary conditions would just have ∇²=0.

What is this 2u (or really any u variable here) and how does one incorporate it into the final solution?

 

[fzero]

Try to solve

u_xx+u_yy=2u

using separation of variables. This is a homogeneous equation. Then f(x,y) is a forcing function for the inhomogeneous equation

u_xx+u_yy - 2u = f(x,y)