How to Show the General Solution to the Poisson Equation?

[yeahhyeahyeah]

Homework Statement

Given that \nabla² 1/r = -4\pi\delta³(r)

show that the solution to the Poisson equation \nabla²\Phi = -(\rho(r)/\epsilon)

can be written:

\Phi(r) = (1/4\pi\epsilon) \int d³r' (\rho(r') / |r - r'|)

Homework Equations

The Attempt at a Solution

I know that the Poisson equation is kind of like a partial differential equation. I rearranged it to \Phi_rr(r²) + \Phi_r(2r) = [-\rho(r) * r² ] / \epsilon

But that wasn't very helpful

Then I also realized that the equations for electric potential is a solution to this... but that is only a special case. Also, is gravitational potential also a solution, or no?

How do you solve this type of equation? What does the 'given': \nabla² 1/r = -4\pi\delta³(r)
even tell me? I am very lost. I read up about Poisson equations and I think the 'given' is like a boundary case... but I don't know how you incorporate the boundary case of a Poisson equation into a solution.

 

[gabbagabbahey]

Just take the Laplacian of the proposed solution (Remember, you don't actually have to solve Poisson's equation to show that something is a solution of it)...what do you get?