Ramanujan–Shen’s differential equations for Eisenstein series of level 2

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), Nidelan (2022) and Toh (2011). As application, we show some arithmetic results on Ramanujan’s tau function.