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For the differential equation $\frac{d^2 y}{d x^2}+k^2 y=0$, 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

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