  In this paper, we study several issues in the linear equation-of-motion (EoM)
and in-in approaches of computing the two-point correlation functions in
multi-field inflation. We prove the equivalence between this EoM approach and
the first-principle in-in formalism. We check this equivalence using several
explicit examples, including cases with scale-invariant corrections and
scale-dependent features. Motivated by the explicit proof, we show that the
usual procedures in these approaches can be extended and applied to some
interesting model categories beyond what has been studied in the literature so
far. These include the density perturbations with strong couplings and
correlated multi-field initial states.
