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Green's Function for Linear ODE

  • #1
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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?
 

Answers and Replies

  • #2
1,847
89
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.
 

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