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In this paper we develop a new approach for studying differential operatorsof an isolated singularity graded hypersurface ring $R$ defining a surface inaffine three-space over a field of characteristic zero. With this method, weconstruct an explicit minimal generating set for the modules of differentialoperators of order two and three, as well as their minimal free resolutions;this expands results of Bernstein, Gel'fand, and Gel'fand and of Vigu\'e. Ourconstruction relies, in part, on a description of these modules that we derivein the singularity category of $R$. Namely, we build explicit matrixfactorizations starting from that of the residue field.

Resolutions of differential operators of low order for an isolated hypersurface singularity