Solving Dirichlet's problem in a rectangle for separate variable coefficient elliptic equations (Q1614108)

The authors consider the problem \[ \begin{cases} U_{xx}(x,y) +{A(x) \over B(y)}U_{yy} (x,y)=0, & (x,y)\in(0,1) \times(0,1),\\ U(x,0)=0, & 0\leq x\leq 1\\ U(0,y)=0,\\ U(x,1)= g(x),\\ U(1,y)=0, \end{cases} \tag{1} \] where \(A(x)\geq 0\), \(B(y)\geq 0\), and \(g(x)\) is a given function. They deal with the construction of exact series solutions of (1) using the separation of variables method.