Structural identifiability is a property of a differential model with parameters that allows for the parameters to be determined from the model equations in the absence of noise. The method of input-output (IO) equations is one method for verifying structural identifiability. This method stands out in its importance because the additional insights it provides can be used to analyze and improve models. However, its complete theoretical grounds and applicability are still to be established. A subtlety and key for this method to work correctly is knowing whether the coefficients of these equations are identifiable. In this article, to address this, we prove identifiability of the coefficients of IO equations for types of differential models that often appear in practice, such as linear models with one output and linear compartment models in which, from each compartment, one can reach either a leak or an input. This shows that checking identifiability via IO equations for these models is legitimate, and as we prove, that the field of identifiable functions is generated by the coefficients of the IO equations. Finally, we exploit a connection between IO equations and the transfer function matrix to show that, for a linear compartment model with an input and strongly connected graph, the field of all identifiable functions is generated by the coefficients of the transfer function matrix even if the initial conditions are generic.