# Green's Function for Linear ODE

## Homework Statement

Find the Green's function for
$$f''(x) + \cos^2 a f(x) = 0;\\ \pm f'(x) + \cos a \cot a f(x)|_{x=x_0(a)}=0$$
where $a$ is a parameter and $x_0$ is defined as
$$x_0(a) = \sec a\arcsin(\cos a)$$.

## Homework Equations

Standard variation of parameters

## The Attempt at a Solution

A solution to the ODE is $$f(x) = \cos(\cos a (x + x_0)) + \cot a \sin\left(\cos a (x + x_0)\right)$$
But this solution satisfies both boundaries. In this case, how do you construct the Green's function since variation of parameters fails?

$$\pm f'(x) + \cos a \cot a f(x)|_{x=\pm x_0(a)}=0$$