On standard bases in rings of differential polynomials

We consider Ollivier's standard bases (also known as differential Gröbner bases) in an ordinary ring of differential polynomials in one indeterminate. We establish a link between these bases and Levi's reduction process. We prove that the ideal $[x^p]$ has a finite standard basis (w.r.t. the so-called $\beta$-orderings) that contains only one element. Various properties of admissible orderings on differential monomials are studied. We bring up the following problem: whether there is a finitely generated differential ideal that does not admit a finite standard basis w.r.t. any ordering.