2D inhomogeneous Laplace equation solution

Question

Hi have an equation that looks like this

$ \begin{equation} \nabla^2\psi(x,y)=w(x,y) \end{equation} $

$w$ is a solution of the homogeneous Laplace equation.

I know that a solution for a 1D inhomogeneous Laplace equation of the form $-\nabla^2U(x)=f(x)$

$ \begin{equation} U(x)=\int_{D}\Phi(x-y)f(y)\,dy \end{equation} $

With $\Phi(x)=-\frac{1}{2\pi}\log|x|$. This I saw in this post.

The exact analytical solution of the integral does not concern me too much because I will solve it numerically, but, there are 2 things I'm concerned with.

1. How do I impose boundary conditions over $\psi(x,y)$ with this solution?

2. Does this approach work for a 2D equation?

The boundary conditions of w are

$ \begin{equation} \begin{split} &w(x,0)=0\\ &w(x,b)=0\\ &w(0,y)=V_1\\ &w(a,y)=V_2\\ \end{split} \end{equation} $

EDIT

The domain is a rectangle centered at 0 that goes until $b$ on $x$ and goes until $a$ on $y$.