Question

Consider Laplace's equation governing the function u(x, y) in the rectangular domain x \in[0, \pi] \text { with } y \in[0,1] u_{x x}+u_{y y}=0 along with the boundary conditions u(0, y)=u_{x}(\pi, y)=0, \text { for } y \in[0,1] \text { with } u_{y}(x, 0)=0 \text { and } u(x, 1)=\frac{1}{2} \text { for } x \in[0, \pi] Use the method of separation of variables to show that the solution satisfying all of the boundary conditions can be written as u(x, y)=\sum_{n=0}^{\infty} \frac{\cosh \left(\left(n+\frac{1}{2}\right) y\right)}{\left(n+\frac{1}{2}\right) \cosh \left(\left(n+\frac{1}{2}\right)\right)} \sin \left(\left(n+\frac{1}{2}\right) x\right) Make sure you explore all possibilities for the domain of the separation constant.

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