Solve Laplace's equation with homogeneous vertical boundary conditions:



From the previous problem we know that the PDE has the solution

$u(x,y) = \displaystyle\sum\limits_{n=1}^\infty\left( A_n \cosh\left(\frac{n\pi}{a}y\right) +B_n \sinh\left(\frac{n\pi}{a}y\right)\right) \sin\left( \frac{n\pi}{a}x\right)$.
with

$\displaystyle A_n =$

----------

$\displaystyle\int$

$dx=$

$B_n=$

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----------

$\displaystyle\int$

$dx -$

$\Biggr)$

$\hskip 15pt =$