This article presents numerical work on a special case of the cosmological semiclassical Einstein equation (SCE). The SCE describes the interaction of relativistic quantum matter by the expected value of the renormalized stress-energy tensor of a quantum field with classical gravity. Here we consider a free massless scalar field with general (not necessarily conformal) coupling to curvature. In a cosmological scenario with flat spatial sections for special choices of the initial conditions, we observe a separation of the dynamics of the quantum degrees of freedom from the dynamics of the scale factor, which extends a classical result by Starobinsky to general coupling. For this new equation of fourth order for the dynamics of the scale factor, we study numerical solutions. Typical solutions show a radiation-like Big Bang for the early universe and de Sitter-like expansion for the late universe. We discuss a specific solution to the cosmological horizon problem that can be produced by tuning parameters in the given equation. Although the model proposed here only contains massless matter, we give a preliminary comparison of the obtained cosmology with the $\Lambda$CDM standard model of cosmology and investigate parameter ranges in which the new models to a certain extent assimilates standard cosmology.