Laplace's equation with insulated vertical edges and when no heat escapes from the lateral faces: $$\displaystyle \frac{\partial^2 u}{\partial x^2} +\frac{\partial^2 u}{\partial y^2}=0, 0< x < a, 0 < y < b$$ $$\displaystyle \frac{\partial u}{\partial x}\Bigg\vert_{x=0}=0, \frac{\partial u}{\partial x}\Bigg\vert_{x=a}=0, 0 < y < b$$ $$\displaystyle u(x,0) = 0, \hskip 10pt u(x,b) = f(x),0<x<a$$
We can solve the PDE using the series solution $u(x,y) = A_0y+ \displaystyle\sum\limits_{n=1}^\infty A_n \sinh\left(\frac{n\pi}{a}y\right) \cos\left( \frac{n\pi}{a}x\right)$. If $f(x) = x$ then

$A_0 =$

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dx

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$A_n =$

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dx

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