As by now you should fully understand from working with the Poisson
equation, one very general way to solve inhomogeneous partial
differential equations (PDEs) is to build a Green's
function^{13.1} and write the solution as an
integral equation.

Let's very quickly review the general concept (for a further discussion don't forget ,). Suppose is a general (second order) linear partial differential operator on e.g. and one wishes to solve the inhomogeneous equation: D f() = &rho#rho;() for .

If one can find a solution
to the associated
differential equation for a *point* source function^{13.2}:
D G(,_0) = &delta#delta;(- _0)
then (subject to various conditions, such as the ability to interchange
the differential operator and the integration) to solution to this
problem is a *Fredholm Integral Equation* (a convolution of the
Green's function with the source terms):
f() = &chi#chi;() + &int#int;_^3 G(,_0)
&rho#rho;(_0) d^3x_0
where
is an *arbitrary* solution to the associated
homogeneous PDE:
D [ &chi#chi;() ]= 0

This solution can easily be verified:
f() & = & &chi#chi;() + &int#int;_^3 G(,_0)
&rho#rho;(_0) d^3x_0

D f() & = & D[ &chi#chi;()] +
D &int#int;_^3 G(,_0) &rho#rho;(_0) d^3x_0

&rho#rho;(_0) d^3x_0

D f() & = & 0 +
&int#int;_^3 D G(,_0) &rho#rho;(_0) d^3x_0

D f() & = & 0 +
&int#int;_^3 &delta#delta;(-_0) &rho#rho;(_0) d^3x_0

D f() & = & &rho#rho;()

It seems, therefore, that we should *thoroughly understand* the ways
of building Green's functions in general for various important PDEs.
I'm uncertain of how much of this to do within these notes, however.
This isn't really ``Electrodynamics'', it is mathematical physics, one
of the fundamental toolsets you need to do Electrodynamics, quantum
mechanics, classical mechanics, and more. So check out Arfken, Wyld,
, and we'll content ourselves with a very quick review of
the principle ones we need: