Laplace Equation Solved by Method of Separation of Variables

[6Stang7]

Homework Statement

Homework Equations

Assume the solution has a form of:

The Attempt at a Solution

It looks like a sine Fourier series except for the 2c₅ term outside of the series, so I'm not sure how to go about solving for the coefficients c₅ and c₁₀. Any idea?

 

[voko]

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.

 

[6Stang7]

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.

Ahhh, so for there to be a term outside the sum due to lambda=0, you must have non-zero values for both X(x) AND Y(y)? Makes sense! :D

 

[LCKurtz]

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 $f$?

 

[6Stang7]

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 $f$?

It is suppose to be u on both sides; that boundary condition is stating: Y'(0)X(x)=Y(0)X(x) since it was assumed the solution to u(x,y) had the form of X(x)Y(y).

 

[LCKurtz]

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.