Upper panels: MZ relation by Peng et al. (2015) (fitted by third-order polynomial functions shown in Fig. 3) for passive and star-forming galaxies. We reproduce it using the new analytical solution presented in this paper and vary different parameter models. In this figure we focus on the timescale parameter τ, and the color code indicates different τ values of the computed galaxies that reside along the MZ relations. In the left panel we adopt a Chabrier (2003) IMF, whereas in the right panel the Salpeter (1955) IMF is assumed. Lower panels: distribution of the predicted passive galaxies (red histogram) and star-forming galaxies (blue histogram) that reproduced the MZ relation in terms of the timescale parameter τ with a Chabrier (2003) IMF (left panel) and a Salpeter (1955) IMF (right panel).