I am currently confronted with a physical equation that, after a fair amount of reworking, can be recast in the form of the modified Mathieu equation :
\begin{equation} y(x)'' - (a - 2q \cosh(2x)) y(x) = 0 \end{equation}
In this case, the parameters $a$ and $q$ are somewhat arbitrary real numbers (q can be any positive real number but is bounded from below at a negative value), although they do obey the relation
\begin{equation} a = k^2 - 2q,\ k \in \mathbb{N} \end{equation}
But from what I can see on the Mathieu equation, those types of situations are very rarely taken into account and are instead generally speaking of the form
\begin{equation} y(x)'' - (a_n(q) - 2q \cosh(2x)) y(x) = 0 \end{equation}
where only certain values of $a$ are considered, with $n$ generally integer, rational or in very rare cases real, and even then usually restricted to specific regions
As far as I can tell, my equation will inevitably cross into the unstable regions for some configurations of the physical system. How can the modified Mathieu equation be dealt with in such circumstances? Does it differ significantly from the Mathieu equation in terms of stability? Is there any exact solution, or if not, can properties still be decided from it?