Here's a question of the Laplace equation in a semi-infinite strip with the Dirichlet Data:
$$ U_{xx}+U_{yy}=0,\qquad 0<x<\infty, \ 0<y<b $$ The boundary conditions are given as $$ u(0,y)=0,u(x,b)=0, u(x,0) = f(x) $$
To solve this problem, we use the Fourier sine transformation with respect to x, and here're the equations and boundary conditions after the transformation:
$$ \frac{d^2}{dy^2}U_s-k^2U_s = 0\qquad [1], \\U_s(k,b) = 0, U_s(k,0) = F_s(k) $$ I'm still confused about how did we obtain the equation [1], and where does $k^2$ come from? Also, how can we find the general solution of the equation [1]? Thanks for the help:)