  Almost commuting operators were introduced in 1985 by George Wilson to
present generalizations the Korteweg-de Vries hierarchy, nowadays known as
Gelfand-Dikey (GD) hierarchies. In this paper, we review the formal
construction of the vector space of almost commuting operators with a given
ordinary differential operator (ODO), with the ultimate goal of obtaining a
basis by computational routines, using the language of differential
polynomials. We use Wilson's results on weigheted ODOs to guarantee the
solvability of the triangular system that allows to compute the homogeneous
almost commuting operator of a given order in the ring of ODOs. As a
consequence the computation of the equations of the GD hierarchies is obtained
without using pseudo-differential operators. The algorithms to calculate the
almost commuting basis and the GD hierarchies in the ring of ODOs are
implemented in SageMath, and explicit examples are provided.
