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For the differential equation $\dfrac{d^2 y}{d x^2}+k^2 y=0$, the boundary conditions are

- $y=0$ for $x=0$, and
- $y=0$ for $x=a$

The form of non-zero solutions of $y$ (where $m$ varies over all integers) are

- $y=\displaystyle{}\sum_m\;\mathrm{~A}_m\;\sin ^{\frac{m \pi x}{a}}$
- $y=\displaystyle{}\sum_m\;\mathrm{~A}_m\;\cos ^{\frac{m \pi x}{a}}$
- $y=\displaystyle{}\sum_m\;\mathrm{~A}_m\;x^{\frac{m \pi}{a}}$
- $y=\displaystyle{}\sum_m\;\mathrm{~A}_{m} \;e^{-\frac{m \pi x}{a}}$