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Here's what I do. Starting with a differential equation:

[tex] a(x) \frac{d^2 y(x)}{dx^2} + b(x) \frac{dy(x)}{dx} +c(x) y(x) = d(x) [/tex]

the green's function solution must satisfy:

[tex] a(x) \frac{d^2 g(x|\xi)}{dx^2} + b(x) \frac{dg(x|\xi)}{dx} +c(x) g(x|\xi) = \delta(x - \xi) [/tex]

Now say we're working in the range [itex]0<x<a[/itex], and the boundary conditions specify either the function or it's first derivative is 0 at each of the endpoints. This means the green's function will satisfy the homogenous DE in the regions [itex]0<x<\xi[/itex] and [itex]\xi<x<a[/itex]. If the homogenous solutions are y

_{1}(x) and y

_{2}(x), then it will have the form:

[tex] g(x|\xi) = \left\{\begin{array}{cc} A_1 y_1(x) + A_2 y_2(x)&0<x<\xi\\B_1 y_1(x) + B_2 y_2(x)&\xi<x<a\end{array} [/tex]

To determine these four coefficients, we get two equations from the boundary conditions, another from requiring it to be continuous, and another from the following equation:

[tex] \lim_{\epsilon \rightarrow 0 } \int_{\xi-\epsilon}^{\xi+\epsilon} \left[a(x) \frac{d^2 g(x|\xi)}{dx^2} + b(x) \frac{dg(x|\xi)}{dx} +c(x) g(x|\xi) \right] dx= \lim_{\epsilon \rightarrow 0 } \int_{\xi-\epsilon}^{\xi+\epsilon} \delta(x - \xi) dx = 1[/tex]

Here's where I'm a little unsure. Do we always get to assume it will be continuous? And when it is continuous, does the above limit always reduce to:

[tex] a(\xi) \left(\frac{d g(x|\xi)}{dx}|_{\xi+} - \frac{d g(x|\xi)}{dx}|_{\xi-} \right) = 1 [/tex]

assuming a(x), b(x), and c(x) are continuous? If this is all right, I have some more questions because some of the solutions I'm getting using it aren't working.