Solve the wave equation with fixed endpoints and the given initial displacement and velocity. $$\displaystyle a^2\frac{\partial^2 u}{\partial x^2} = \frac{\partial^2 u}{\partial t^2}, 0< x < \pi, t > 0$$ $$\displaystyle u(0,t) = 0, u(\pi,t) = 0, t > 0$$ $$\displaystyle u(x,0) = \frac{1}{6}x(\pi^2-x^2)$$ $$\displaystyle\hskip 10pt \frac{\partial u}{\partial t}\Bigg\vert_{t=0}=0,0<x<\pi$$

Now we can solve the PDE using the series solution $u(x,t) = \displaystyle\sum\limits_{n=1}^\infty \Bigg( A_n$ $+ B_n$ $\Bigg)$. The coefficients $A_n$ and $B_n$ are Fourier coefficients we have computed many times:

$A_n$

$=$

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$\displaystyle\int$

$dx$

$\hskip 12pt$

$=$

$B_n$

$=$

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$\displaystyle\int$

$dx$

$\hskip 12pt$

$=$