Understanding Greens Function vs. Variation of Parameters

[Xyius]

So I just recently learned about how to use Greens Function to solve a differential equation. The formula was derived and it said the main goal was to find an integral representation to the solution. It seems to me, however that Greens Function is nothing more than variation of parameters with a different label attached to it. Is their a point to Greens Function? Why use greens function when you can use variation of parameters? (They seem like they are the exact same thing.)

Any help in understanding this would be appreciated! :D

 

[Xyius]

Oops! I just realized, this should be in the Differential equations section.

 

[HallsofIvy]

I have moved it.

One point to Green's function is that it gives of a way of writing and talking about the solution to a differential equation without actually solving it:
The general solution to the non-homogenous linear differential equation L(y)= f(x) (where L is some linear differential operator) is
$$y(x)= \int G(x,t)f(t)dt$$
where G is the Greens function corresponding to L.

Advanced Physics texts often give solutions to complicated equations in terms of the Green's function, without specifying what the Green's function is, just to be able to talk about a solution to a differential problem that is too difficult to actually solve in a reasonable time.

 

[AlephZero]

Most differential equations can not be solved in a closed form (except by inventing a new "special function" that just happens to be the solution, like Bessel functions, Fresnel integrals, etc).

However you may be able to integrate the Green's function solution for a particular value of x (or the limit as x goes to infinity, or whatever) without knowing the general solution, or you may be able to evaluate the integral numerically. In fact this is a general way to create finite element approximations for solving ODEs and PDEs.

 

[Xyius]

Ohh okay! I think I understand. Thanks a lot :)