# Solving a 2nd order ODE using Green's Function

ultimateguy

## Homework Statement

The homogeneous Helmholtz equation

$$\bigtriangledown^2\psi+\lambda^2\psi=0$$

has eigenvalues $$\lambda^2_i$$ and eigenfunctions $$\psi_i$$. Show that the corresponding Green's function that satisfies

$$\bigtriangledown^2 G(\vec{r}_1, \vec{r}_2)+\lambda^2 G(\vec{r}_1, \vec{r}_2)=-\delta(\vec{r}_1-\vec{r}_2)$$

may be written as

$$G(\vec{r}_1, \vec{r}_2)=\sum_{i=1}^{\infty}\frac{\psi_i(\vec{r}_1)\psi_i(\vec{r}_2)}{\lambda^2_i-\lambda^2}$$

## Homework Equations

$$\int(\psi \bigtriangledown^2 G-G\bigtriangledown^2 \psi) d\tau_2=\int(\psi \bigtriangledown G-G\bigtriangledown\psi) d\sigma$$

## The Attempt at a Solution

I'm using Arfken's Mathematical methods for physicists, and it isn't very good at explaining the examples it uses. I just need some kind of jump start to get me going. Do I need to use the equation in the relevant equations section?