Solve Laplace's equation with homogeneous vertical boundary conditions:

\displaystyle
\frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2}=0, 0< x < a, 0 < y < b

\displaystyle
u(0,y)=0, \hskip 10pt u(a,y)=0, 0 < y < b

\displaystyle u(x,0) = 1, \hskip 10pt u(x,b) = x,0<x<a

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

\hskip 15pt =

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

with

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