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## Homework Statement

The steady state temperature distribution, T(x,y), in a flat metal sheet obeys the partial differential equation:

[tex]\frac{\partial^2{T}}{\partial{x}^2}+{\frac{\partial^2{T}}{\partial{y}^2}}=0[/tex]

Separate the variables and find T everywhere on a square flat plate of sides S with boundary conditions:

[tex]T(0,y)=T(S,y)=T(x,0)=0[/tex]

[tex]T(x,S)=T_0[/tex]

## Homework Equations

## The Attempt at a Solution

For a solution, I get:

[tex]T(x,y)=\sum_{n=1}^{\infty}\frac{4T_0}{n\pi}sin(\frac{n\pi}{S}x)\frac{sinh(\frac{n\pi}{S}y)}{sinh(n\pi)}[/tex]

I am not sure if I have given a sufficient enough answer for the coefficient of the series. I got the coefficient by doing the following:

[tex]T(x,s)=\sum_{n=1}^{\infty}A_n{sin(\frac{n\pi}{S}x)}sinh(n\pi)=T_0[/tex]

[tex]B_n=A_n{sinh(n\pi)}[/tex]

[tex]B_n=\frac{2}{S}\int_{0}^{S}T_0{sin(\frac{n\pi}{S}x)}dx[/tex]

[tex]B_n=\frac{4{T_0}}{n\pi}[/tex] for "n" odd and 0 for "n" even.

Does anyone know if the coefficient is incorrect?