J. K. Zhou introduced what some authors call the differential transform method (DTM) cf. Differential transformations and its application for electrical circuits (Chinese), Huazhong University Press, Wuhan, China (1986)]. The idea behind this notion is to assume analyticity and to use Taylor series expansion to determine the solution. The author states several properties of the DTM transformation in the one-dimensional case, i.e., properties of Taylor coefficients, in a purely formal way.
Similar statements can be found in the two-dimensional setting [cf. F. Ayaz, Appl. Math. Comput. 147, No. 2, 547–567 (2004; Zbl 1032.35011), C. K. Chen and S. H. Ho, Appl. Math. Comput. 106, No. 2–3, 171–179 (1999; Zbl 1028.35008)]. Three examples are calculated at the end of the paper.