The problem I'm working on asks to use separation of variables to derive a solution of the Dirichlet problem for the right quarter-plane $x>0,y>0$ if $u(x,0)=f(x)$ for $x>0$ and $u(0,y)=g(y)$ for $y>0.$
I carried out the separation of variables and obtained $$X_w = a_w\cos(wx)+b_w\sin(wx)$$
using the boundary condition $u(0,y)=g(y)$, I obtained $$X(0)=a_w\cos(0)+b_w\sin(0)$$ $$g(y)=a_w $$
I've never had a problem where the constant equals a function, so I'm not sure how to continue from here.
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rudytheduck
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