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If we take $\delta = 1$, we have $u'' + (1 + \epsilon\cos 2t)u = 0$. I need to find a $u(t) = u_1(t) + u_2(t)$ such that it satisfies $u_1(t) = 1$, $u_1'(t) = 0$, $u_2(t) = 0$, $u_2'(t) = 1$ and $u$ satisfies $u'' + (1 + \epsilon\cos 2t)u = 0$. Since $\epsilon\ll 1$, can we make $\epsilon = 0$? Because if that was the case, then $u = A\cos t\sqrt{\delta} + B\sin t\sqrt{\delta}$.For Mathieu's equation
$$
u'' + (\delta + \epsilon\cos 2t)u = 0,
$$
use the perturbation method to analytically determine its stability near $\delta = 1$ and $\delta = 4$ when $\epsilon\ll 1$.
How would I do this using a perturbation method?