Differential standard bases under composition

We generalize Hoon Hong's theorem on Gröbner bases under composition to the case of differential standard bases in the ordinary ring of differential polynomials $\mathcal F\{y\}$. In particular, we prove that some ideals have finite differential standard bases. We construct special orderings on differential monomials such that ideals generated by some power of a quasi-linear polynomial acquire finite differential standard bases.