Time-dependent Stokes shift (TDSS) measurements provide crucial insights into the dynamics of liquids. The interpretation of TDSS measurements is often aided by molecular dynamics simulations, where solvent response functions are computed either with an equilibrium or nonequilibrium approach. In the nonequilibrium approach, the solvent is at equilibrium with the ground electronic state of the solute and its charge distribution is instantaneously changed to that of the first excited state. The solvation response function is then calculated as a nonequilibrium average of the subsequent evolution of the solvent influence on the electronic energy gap. In the equilibrium approach, the normalized time correlation function of the fluctuations of the solvent-perturbed electronic energy gap is calculated. If the linear response approximation is valid, then the nonequilibrium solvation response function is identical to the equilibrium time correlation function. The nonequilibrium methodology conceptually mimics the experiment, but it is significantly more computationally expensive than the equilibrium approach. In multicomponent systems such as ionic liquids, it is natural to inquire how the various components affect the observed relaxation dynamics. When utilizing the nonequilibrium methodology, the solvation response naturally decomposes into a sum of responses for each component present in the system. However, the equilibrium time correlation function does not decompose unambiguously. Here, we have evaluated a decomposition strategy that is consistent with the linear response approximation for the study of solvation dynamics of coumann 153 (C153) in the 1-ethyl-3-methyl imidazolium tetrafluoroborate, [emim][BF4], ionic liquid. The agreement of the equilibrium and nonequilibnum solvation response functions demonstrates the validity of the linear response approximation for the C153/[emim] [BF4] system. Moreover, decompositions of the equilibrium time correlation function into contributions of the translational and rovibrational motions of the anions and cations are essentially identical to the same decompositions of the nonlinear solvation response.