A general result is proven concerning time-variant displacement equations with positive solutions in a general operatorial setting. It is then shown that the solutions of several completion problems, recently considered in connection with classical interpolation and moment theory, follow as special cases of the main result. The main purpose of this paper is to show that under supplementary finite-dimensionality conditions, a so-called generalized Schur algorithm, which naturally arises in connection with displacement equations, can be used to prove the above-mentioned result. The associated transmission-line interpretation is also discussed in terms of a cascade of elementary sections with intrinsic blocking properties.