The Mathieu equation has periodic solutions and therefore it is possible to use finite difference approximations to obtain an equivalent matrix eigenvalue problem. The matrix will of tridiagonal kind with an extra two coefficients, one top right and the other bottom left when second order accurate differences are used.

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Mathiew's differential equation is a second order differential equation of the form d2y/dx2 + (a -2qcosx)y = 0 . For that you have first to transform the second order equation into a first order differential system dy/dx = z and dz/dx = (a-2qcosx)y , second use a Runge-Kutta.

This link has more info https://en.wikipedia.org/wiki/Mathieu_function