Status of the differential transformation method

Further to a recent controversy on whether or not the differential transformation method (DTM) for solving a differential equation is purely and solely the traditional Taylor series method, it is emphasized that the DTM is currently used as a technique for calculating the power series of the solution (in terms of the initial value parameters) including, sometimes, a piecewise analytic continuation process which is implemented either in a numerical routine (e.g. within a shooting method) or in a semi-analytical procedure (e.g., to solve a boundary value problem). Emphasized also is the fact that, at the time of its invention, the currently-used basical ingredients of the DTM (that transform a differential equation into a difference equation of same order that is iteratively solvable) were already known for a long time by the Taylor-method users (notably in the elaboration of numerical routines for automatically solving ordinary differential equations). The procedure is still utilized in perfect indifference of the defenders of the DTM (and reciprocally). Conversely, the DTM is much formalized, set on the same footing as the Fourier, Laplace or Mellin transformations, easily adaptable to different kinds of differentiation procedures. That has made it very attractive. Hence a systematic application to various problems of the Taylor method, and more generally of the power series method (including noninteger powers) has been made possible. It seems that its potential has not been exploited as it could be. After a discussion on the reasons of the misunderstandings which have caused the controversy, the preceding topics are concretely illustrated. It is concluded that, when the DTM is applied to ODEs, it should be mentioned that the DTM coincides with the traditional Taylor method, contrary to what is currently written.