Green's Function for Linear ODE

[member 428835]

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?

 

[member 428835]

I think someone commented but then deleted their comment, because I received an email but now cannot see their response. Anyways, I mistyped the boundaries, which are at each endpoint:
$$
\pm f'(x) + \cos a \cot a f(x)|_{x=\pm x_0(a)}=0$$

Sorry for the confusion.