Say we have a second order linear differential equation, $$ y''(t)+a y(t)=0,$$ where $a$ is constant with respect to the independent variable $t$.
Now, if we have a periodic (with period $T$) boundary condition to be applied on the solutions of this equation, $y(t)$, such as $$ y(t_{0}+T)=y(t_{0}), $$ is there a way to directly incorporate this periodic boundary condition into the differential equation itself, so that it then contains both the original equation above and the boundary?
For example, looking at the Mathieu equation of the form $$y''+(a+ 2 \epsilon \cos t)y=0,$$ where $\epsilon$ is another parameter, I find it tempting to ask this question, because this equation seems to already include such periodicity automatically (the periodic boundary becomes periodic coefficient in the equation now?).
But I am not sure how accurate this way of thinking is. Any advice would be appreciated.