  The density matrix formalism and the equation of motion approach are two
semi-analytical methods that can be used to compute the non-equilibrium
dynamics of correlated systems. While for a bilinear Hamiltonian both
formalisms yield the exact result, for any non-bilinear Hamiltonian a
truncation is necessary. Due to the fact that the commonly used truncation
schemes differ for these two methods, the accuracy of the obtained results
depends significantly on the chosen approach. In this paper, both formalisms
are applied to the quantum Rabi model. This allows us to compare the
approximate results and the exact dynamics of the system and enables us to
discuss the accuracy of the approximations as well as the advantages and the
disadvantages of both methods. It is shown to which extent the results fulfill
physical requirements for the observables and which properties of the methods
lead to unphysical results.
