Mathieu equation solution with non-periodic boundary conditions

Question

I need to solve the Mathieu equation:

$y''(x)+(a-2q \cos(2x)) y(x) = 0$

but with the unusal boundary condition:

$y(x+\pi) = e^{i \alpha}y(x) \quad , \quad \alpha \in R$

if $\alpha = 0$ than the solution are $se_n , ce_n$, but what about this case?

thanks