(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

## \frac{\partial^2 V}{\partial x^2} +\frac{\partial^2 V}{\partial y^2} = 0 ##

it's defined for ##V=0 ## for

## x=0, 0<y<b##,

## x=a, 0<y<b##,

## y=0, 0<x<a##

##V=V_0 x/a ## for ## y=b, 0<x<a##

2. Relevant equations

None.

3. The attempt at a solution

I've got

##V(x,b) = \displaystyle V_0 x/a = \sum_{n=1}^{\infty} K_n \sin ({n \pi x \over a}) \sinh ({n \pi b \over a}) ##

At first I thought I could set K to 0 for n>1 and do a simple division, but I don't know what K1 is. Do I always need to find Fourier coefficients. I have seen a problem www.robots.ox.ac.uk/~jmb/lectures/pdelecture5.pdf here where in the final stage they do something like this. I'm guessing I need to do the former, but I am interested to know if the other method is possible.

As for the coefficients, the function is odd so the only integral I need is the sine integral.

## K_n = \frac{1}{a} \int_0^{2a} \frac{V x}{a} \sin ({n \pi x \over a}) dx =-\frac{2V}{n\pi} ##

I think this is right:

##V(x,y) = \sum_{n=1}^{\infty} -\frac{2V}{n\pi} \sin ({n \pi x \over a}) \sinh ({n \pi y \over a})##

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# Homework Help: Solving the 2D laplace equation. Quick question about the final stage

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