Green’s functions: the genius way to solve DEs

Green’s functions is a very powerful and clever technique to solve many differential equations, and since differential equations are the language of lots of physics, including both classical mechanics, electrodynamics, and even quantum field theory, so it is important to know how it works. Of course, this includes some explanation and perhaps a pretty different motivation for Dirac delta function, which is pretty weird, but also not really when you think about it a different way.

They don’t state the boundary / initial conditions explicitly, and they don’t even use x and xi, or t and tau, usually just their difference. Usually it is that the Green’s functions vanish when the position is far away from the origin, and for those involving time, 0 before time tau, assuming that tau is greater than 0 (the so-called “advanced” Green’s function)