The Dirichlet problem for a mixed-type equation with characteristic degeneration in a rectangular domain

We study the first boundary problem for the following mixed-type equation of the second kind:
$$ u_{xx}+yu_{yy}+au_y-b^2u=0 $$
in the domain $\{(x,y)\mid0<x<1,\ -\alpha<y<\beta\}$, where $a,b,\alpha$, and $\beta$ are given real numbers, and $0<a<1$, $b\geq0$, $\alpha>0$, $\beta>0$. Based on the completeness of the system of eigenfunctions of one-dimensional spectral problem we establish a uniqueness criterion. We construct a solution to the problem as the sum of the series in eigenfunctions.