Solve



using superposition $u = u_1+u_2$ and

$u_1(x,y) = \displaystyle\sum\limits_{n=1}^\infty \left( A_n \cosh\left(\frac{n\pi}{a}y\right)+B_n \sinh\left(\frac{n\pi}{a}y\right) \right) \sin\left( \frac{n\pi}{a}x\right)$
$u_2(x,y) = \displaystyle\sum\limits_{n=1}^\infty \left( A_n^\prime \cosh\left(\frac{n\pi}{b}x\right)+B_n^\prime \sinh\left(\frac{n\pi}{b}x\right) \right) \sin\left( \frac{n\pi}{b}y\right)$

$\displaystyle A_n =$

$\hskip 20pt$ ---------------------------

$\displaystyle\int$

$dx=$

$\displaystyle A_n^\prime =$

$\hskip 20pt$ ---------------------------

$\displaystyle\int$

$dy=$

$B_n=$

$\hskip 40pt 1$ -----------------------------------

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----------

$\displaystyle\int$

$dx -$

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$\hskip 15pt =$

$B_n^\prime=$

$\hskip 40pt 1$ -----------------------------------

$\Biggl($

----------

$\displaystyle\int$

$dy -$

$\Biggr)$

$\hskip 15pt =$