A reduction of certain analytic differential equations to differential equations of an algebraic type

Abstract

The theorem that a system of power series in n dependent variables and p independent variables can be reduced to an equivalent system which are polynomials with respect to the dependent variablest leads rather naturally to the suggestion that possibly a corresponding reduction can be effected in the case of analytic differential equations. It might be anticipated that the theory for such a reduction of differential equations is a more complicated one than the corresponding theory for implicit functions. In certain respects this anticipation is amply justified, while in certain other respects, perhaps, it is not. In effecting such a reduction many different methods and points of view are possible. The method adopted in the present paper is a substitution on the dependent variables of the form