Poisson's equation, maybe Green function

fluidistic

1. The problem statement, all variables and given/known data
Calculate the solution to [itex]\triangle u (r, \phi )=1[/itex] in a circle of radius a with u(r=a)=0.


2. Relevant equations
Green function I think, the exercise is listed with other exercises related to Green function. So even though separation of variables works here or any other method, I'd prefer to solve it via Green's function method.


3. The attempt at a solution
I'm self studying Green function and having an extremely hard time to get any intuition on this.
Let's call D the Laplacian operator. I have that [itex]Du=1=g[/itex].
I think that the Green function depends only on the linear operator D and never on g. The Green function related to the Laplacian is, I believe, [itex]DG(r,t)=\frac{1}{4 \pi |r|}[/itex] or something similar to this that I don't understand. I don't even know if this is true when the Laplacian is in polar coordinates.
I wish someone could explain me how to tackle this problem, I'm willing to put a lot of efforts in it.