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Differential Equations

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Moderate

Forcing Term Consisting of Different Frequencies

DIFFEQ-LOVZAI

Consider:

$$y''=-4k^2y+\sum_{n=1}^{\infty}b_n\cos(n\gamma x)$$

...where:

$$k\neq 0$$
$$\gamma>0$$
$$\sum_{n=1}^{\infty}|b_n|<\infty$$

Define $\omega=2|k|$, and assume $\omega\neq n\gamma$ for any $n\in\mathbb{Z}$. The full list of solutions is

A

$C_1e^{\omega x}+C_2e^{-\omega x}+\sum_{n=1}^{\infty} \cfrac{b_n}{4k^2-n^2\gamma^2}\cos(n\gamma x)$

B

$C_1\cos(\omega x)+C_2\sin(\omega x)+\sum_{n=1}^{\infty} \cfrac{b_n}{\omega^2-n^2\gamma^2}\cos(n\gamma x)$

C

$C_1\cos(\omega x)+C_2\sin(\omega x)+\sum_{n=1}^{\infty}b_n\cos(n\gamma x)$

D

None of the above