I am working on a problem which requires the sign of the Green's function of Mathieu equation with periodic boundary conditions.
So, if we have
$$x'' + \left( p + 2q \cos 2t \right) x = 0$$
with the boundary conditions $x \left( 0 \right) = x \left( 2 \pi \right)$ and $x' \left( 0 \right) = x' \left( 2 \pi \right)$.
We know that $p$ is not an eigenvalue so that the only solution satisfying the bundary conditions is the zero (trivial) solution. Now, the general method to construct Green's function would suggest us,
$$G \left( t, s \right) = \begin{cases} \dfrac{y_1 \left( t \right) y_2 \left( s \right)}{W \left( y_1, y_2 \right)} & t < s \\ \dfrac{y_1 \left( s \right) y_2 \left( t \right)}{W \left( y_1, y_2 \right)} & s < t \end{cases}$$
where $y_1, y_2$ are the Mathieu sine and cosine functions respectively and $W$ is the Wronskian.
However, since $p$ is not an eigenvalue, the functions $y_1, y_2$ are not periodic so that the Green's function does not satisfy the boundary conditions. Where have I gone wrong?
Also, if this method is not applicable here, then is there any way to check for the sign of Green's function or the sign of the integral $\int\limits_{0}^{2 \pi} G \left( t, s \right) \ ds$?