Identifiability is a structural property of any ODE model characterized by a set of unknown parameters. It describes the possibility of determining the values of these parameters from fusing the observations of the system inputs and outputs. This paper finds the general analytical solution of this fundamental problem and, based on this, provides a general and automated analytical method to determine the identifiability of the unknown parameters. In particular, the method can handle any model, regardless of its complexity and type of non-linearity, and provides the identifiability of the parameters even when they are time-varying. In addition, it is automatic as it simply needs to follow the steps of a systematic procedure that only requires to perform the calculation of derivatives and matrix ranks. Time-varying parameters are treated as unknown inputs and their identification is based on the very recent analytical solution of the unknown input observability problem [1, 2]. The method is used to determine the identifiability of the unknown time-varying parameters that characterize two non-linear models in the field of viral dynamics (HIV and Covid-19) and a non-linear model that characterizes the genetic toggle switch. New fundamental properties that characterize these models are determined and discussed in detail through a comparison with the state-of-the-art results. In particular, regarding the very popular HIV ODE model and the genetic toggle switch model, the method automatically finds new important results that are in contrast with the results in the current literature.