I have faced with the following problem. I have a potential well at $[-1,1]$ and trying to solve eigenfunctions and eigenvalues problem: $$N'' + \alpha^2 N = 0, \quad N'|_{x = 0} = 0, \quad N|_{x = 1} = 0,$$ where $\alpha$ is defined by problem conditions.
Ok, that's easy: $$N_p = A_p\cos(\alpha_px), \quad \alpha_p = \frac{\pi}{2} + \pi\cdot p, \quad p = 0,1,2...$$ This is first approximation.
Then, I need to apply some kind of external force $f(x) = B\cos(\alpha_0^1x),$ where $\alpha_0^1$ is eigenvalue from first approximation, corresponding to $p = 0$, i.e. $\alpha_0^1 = \frac{\pi}{2}$.
What are the eigenvalues and eigenfunctions now for the equation; $$N'' + \alpha^2 N = B\cos(\alpha_0^1x).$$
I obtained, that $$N_p = K\cos(\alpha_p) + \dfrac{B\cos(\alpha_0^1x)}{\alpha_p^2 - (\alpha_0^1)^2}, \quad p = 1, 2, 3...$$.
But then there is a resonance as $\alpha = \alpha_0^1$. What I need to do to solve this problem? Thank you!