Solve Laplaces equation inside a rectangle 0 x L 0 y H with

Solve Laplace’s equation inside a rectangle 0 x L, 0 y H, with the following boundary conditions [Hint: Separate variables. If there are two homogeneous boundary conditions in y, let u(x,y)=h(x)(y), and if there are two homogeneous boundary conditions in x, let u(x,y)=(x)h(y).]:

(c) u/x (0,y) = 0, u(L,y)=g(y), u(x,0) = 0, u(x,H)=0

The final result must be

u(x,y) = (from n=1 to ) A_n cosh(nx /L) sin(nx /L)

where

A_n cosh(nx /L) =2/H ( from 0 to H) g(y) sin(nx /L) dy

Could you help me please because I am very stuck.

Thank you in advance

Solution

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https://drive.google.com/open?id=0B1j-Nlh5wh4XNDN0MGotdTdfRms