Characterization of PDE Reducible to ODE under a Certain Homogeneity and Applications to Singular Cauchy Problems

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.