Green's Function

To construct the Green's function for this system, find any two solutions, say $w (x,\lambda)$ and $z(x,\lambda)$ , which satisfy the two respective boundary conditions:

$$w(x,\lambda) = \cos \sqrt{\lambda} x~~,$$

$$z(x,\lambda) = \cos \sqrt{\lambda} (x-\ell)~~.$$

The Green's function is

$$G_\lambda (x;\xi) = \frac{-1}{c} w(x_<,\lambda)z(x_>,\lambda)$$

$$= \frac{-1}{c(\lambda)} \left\{ \begin{array}{cc} \cos \sqrt{\l... ...\cos \sqrt{\lambda} (x -\ell) &\xi \le x ~~. \end{array}\right.$$

Here

$$c(\lambda) = p(x)~\left( w(x)z'(x)~-~w'(x)z(x) \right)$$

$$~ = ~~1~~\left( \cos \sqrt{\lambda} x ~(-)\sqrt{\lambda}\sin \sqrt{\l... ...ell) ~~-~~\sqrt{\lambda}\sin \sqrt{\lambda}x \cos\sqrt{\lambda}(x-\ell) \right)$$

$$= ~~ -\sqrt{\lambda}~~ \sin \sqrt{\lambda} (x-\ell -x) ~=~\sqrt{\lambda}\sin \sqrt{\lambda}\ell$$

Thus

$$G_\lambda (x;\xi)=\left\{ \begin{array}{cc} \displaystyle - ... ...da}\sin \sqrt{\lambda}\ell } &\xi \le x ~~. \end{array}\right.$$