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Laplace Equation Solved by Method of Separation of Variables 
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#1
Nov1212, 08:36 PM

P: 213

1. The problem statement, all variables and given/known data
2. Relevant equations Assume the solution has a form of: 3. The attempt at a solution It looks like a sine Fourier series except for the 2c_{5} term outside of the series, so I'm not sure how to go about solving for the coefficients c_{5} and c_{10}. Any idea? 


#2
Nov1312, 06:56 AM

Thanks
P: 5,801

When lambda is zero, X(x) is identically zero, which means X(x)Y(y) is also zero. So there must be nothing in front of the series.



#3
Nov1312, 08:45 PM

P: 213




#4
Nov1312, 09:24 PM

HW Helper
Thanks
PF Gold
P: 7,665

Laplace Equation Solved by Method of Separation of Variables
I am puzzled by the boundary condition$$
\left. \frac{\partial u}{\partial y}\right _{y=0} = u(x,0)$$Is that supposed to be the same ##u## on both sides? Or is it just another way to say something like$$ u_y(x,0) = f(x)$$some arbitrary function ##f##? 


#5
Nov1412, 11:49 PM

P: 213




#6
Nov1512, 11:17 AM

HW Helper
Thanks
PF Gold
P: 7,665

OK. With that clarification for me, I would just comment about the last three lines. You already know you should have no eigenfunction for ##\lambda = 0##. Your eigenvalues are ##\lambda_n = n\pi##. Your third line from the bottom should read for the eigenfunctions ##Y_n##$$
Y_n(y) = n\pi\cosh(n\pi y)+\sinh(n\pi y)$$You don't need a constant multiple in front of them and there shouldn't be an ##x## in front of the ##\cosh## term. Similarly your eigenfunctions for ##X## are$$ X_n(x) = \sin(n\pi x)$$ Then you write your potential solution as$$ u(x,y) =\sum_{n=1}^\infty c_nX_n(x)Y_n(y)= \sum_{n=1}^\infty c_n\sin(n\pi x)(n\pi\cosh(n\pi y)+\sinh(n\pi y))$$Now you are ready for the Fourier Series solution to the last boundary condition. 


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