So after using the method of separation of variables of a PDE in 3 variables, I have simplified it to a ODE in time and a PDE of two variables. However, my resulting PDE is a linear second order partial differential equation with variable coefficients of the form
$Au_{xx} + Bu_{yy} + C(x)u_x + D(y)u_y + (p(x,y) + E)u = 0$
where A,B, and E are constants, p(x,y) is a polynomial of x and y of degree 2, and C(x) and D(y) are functions of x and y respectively. Any suggestions on how to solve the resulting PDE?