Differential equations satisfied by Eisenstein series of level 2

Ramanujan’s differential equations for the classical Eisenstein series are of great importance to many areas in number theory and special functions. H.H. Chan recently demonstrated that these differential equations can be derived from the triple product identity and the quintuple product identity in an elementary manner. In this article, we extend this method in a uniform manner to derive corresponding differential equations for the Eisenstein series of level 2. Several applications of these differential equations are also given.