Local uniqueness of a boundary value problem in the complex plane

We will consider the following boundary value problem for a second-order linear differential equation with polynomial coefficients in the complex plane: is there a differential equation of the specific type with linearly independent solutions y₁(x) and y₂(x) such that the ratio y₁(x)/y₂(x) can be prescribed both at the origin and at infinity? We shall derive uniqueness conditions for the problem.