MR3114822 Takuya WATANABE and Jiichiroh URABE Takuya WATANABE and Jiichiroh URABE Characterization of PDE Reducible to ODE under a Certain Homogeneity and Applications to Singular Cauchy Problems Funkcialaj Ekvacioj. Serio Internacia 56 2013 225--247 http://fe.math.kobe-u.ac.jp/FE/FullPapers/56-2/56_225.pdf http://www.ams.org/mathscinet-getitem?mr=MR3114822 We give a necessary and sufficient condition for a homogeneous partial differential equation in two variables to be reduced to a homogeneous ordinary one under a certain change of variables. It is described by means of the commutator with a first order partial differential operator which characterizes a homogeneity. Moreover we obtain the explicit representation of the reduced ordinary differential equation. This result is a generalization of such a reduction which had been applied to singular Cauchy problems in our previous works [U, WU1]. This fact suggests that local structures of the solutions to partial differential equations can be described by global structures of those to ordinary ones. Commutation relation, Euler's homogeneous function theorem, A singular Cauchy problem, Hypergeometric differential equation. 34A25, 35A20, 35G05, 44A45. 56-225 2013 Characterization of PDE Reducible to ODE under a Certain Homogeneity and Applications to Singular Cauchy Problems Takuya WATANABE and Jiichiroh URABE Takuya WATANABE and Jiichiroh URABE
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