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I do not know how to begin with the following problem:

you are asked to solve the wave equation subject to the boundary conditions ($u(0,t)=u(L,t)=0$), $u(x,0)=f(x)$ for $0\le x\le L$ and ${u}_{t}(x,0)=g(x)$ for $0\le x\le L$ . Hint: using the $u(x,t)=\sum_{n=1}^{\infty}{{b}_{n}\sin(\frac{n\pi x}{L})\cos(\frac{n\pi c t}{L})}$ and the remark is that if the initial velocity is nonzero, then additional terms of the form ${{{b}^{*}}_{n}\sin(\frac{n\pi x}{L})\cos(\frac{n\pi c t}{L})}$ must be included where n is nonnegative integer.

$f(x)=\sin{\frac{2\pi x}{L}}$ and $g(x)=0$

Thanks,

Cbarker1