Some identities related to Eulerian polynomials and involving the Stirling numbers

In the paper, the authors establish two identities, which can be regarded as nonlinear differential equations, for the generating function of Eulerian polynomials, find two identities for the Stirling numbers of the second kind, present two identities for Eulerian polynomials and higher order Eulerian polynomials, and pose two open problems about summability of two finite sums involving the Stirling numbers of the second kind. Some of these conclusions meaningfully and significantly simplify several known results.