  Ramanujan (1916) and Shen (1999) discovered differential equations for
classical Eisenstein series. Motivated by them, we derive new differential
equations for Eisenstein series of level 2 from the second kind of Jacobi theta
function. This gives a new characterization of a system of differential
equations by Ablowitz-Chakravarty-Hahn (2006), Hahn (2008), Kaneko-Koike
(2003), Maier (2011) and Toh (2011). As application, we show some arithmetic
results on Ramanujan's tau function.
