How to Show the General Solution to the Poisson Equation?

[yeahhyeahyeah]

Homework Statement

Given that $$\nabla$$² 1/r = -4$$\pi$$$$\delta$$³(r)

show that the solution to the Poisson equation $$\nabla$$²$$\Phi$$ = -($$\rho$$(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': $$\nabla$$² 1/r = -4$$\pi$$$$\delta$$³(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?