Solving ODE using Green's function

Question

How do I solve ODEs for the Green's function, for the given homogeneous boundary or initial conditions:

$$G″ - \kappa_0^22 G(x, x_0) = \delta(x - x_0),\quad G = 0\text{ for }x = a\text{ and }x = b,\quad\kappa_0\in\mathbb{R}$$

Answer 1

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I'll assume that

$\ds{\partiald[2]{\mrm{G}\pars{x,x_{0}}}{x} - k_{0}^{2}\,\mrm{G}\pars{x,x_{0}} = \delta\pars{x - x_{0}}\,,\qquad \left\{\begin{array}{l} \ds{\mrm{G}\pars{a,x_{0}} = \,\mrm{G}\pars{b,x_{0}} = 0} \\[2mm] \ds{k_{0} > 0\quad \mbox{and}\quad x_{0} \in \pars{a,b}} \end{array}\right.}$.

\begin{align} \mrm{G}\pars{x,x_{0}} & = \left\{\begin{array}{lcl} \ds{A\sinh\pars{k_{0}\bracks{x - a}}} & \mbox{if} & \ds{x < x_{0}} \\[2mm] \ds{B\sinh\pars{k_{0}\bracks{b - x}}} & \mbox{if} & \ds{x > x_{0}} \end{array}\right. \end{align} $\ds{A}$ and $\ds{B}$ are determined with:

1. $\ds{\mrm{G}\pars{x,x_{0}}}$ is continuos at $\ds{x = x_{0}}$.
2. $\ds{\mrm{G}\pars{x,x_{0}}}$ satisfies $\ds{\left.\partiald{\mrm{G}\pars{x,x_{0}}}{x} \right\vert_{\ x\ =\ x_{0}^{-}}^{\ x\ =\ x_{0}^{+}}\,\,\, = 1}$

Namely, $$ \left\{\begin{array}{rcrcl} \ds{\sinh\pars{k_{0}\bracks{x_{0} - a}}\, A} & \ds{-} & \ds{\sinh\pars{k_{0}\bracks{b - x_{0}}}\, B} & \ds{=} & \ds{0} \\[2mm] \ds{-k_{0}\cosh\pars{k_{0}\bracks{x_{0} - a}}\, A} & \ds{-} & \ds{k_{0}\cosh\pars{k_{0}\bracks{b - x_{0}}}\, B} & \ds{=} & \ds{1} \end{array}\right. $$