Linear Partial Differential Equations with Constant Coefficients: an Elementary Proof of an Existence Theorem

Abstract

and any solution of R(DV ...,Dm)z = 0 (3) is also a solution of (1). The converse proposition, that every integral of (1) is the sum of an integral of (2) and an integral of (3), was postulated by Hadamard (1), in the case of m = 2, for linear equations with constant or variable coefficients, provided only that the two operators Q, R are commutative. This result was established by Cerf (2) and by Janet (3), who extended it to a very general case which certainly includes that under consideration here. The proof of the general theorem is not simple, however ; and in the case mentioned below (§3), in which the equation is fully reducible, most textbooks are content to assume the result without proof. We shall now give a purely elementary proof of this converse theorem, in the case when one of the factors of P, R say, is a power of a linear expression in Dv ..., Dm, which is not a factor of Q. We shall make the hypothesis that any partial differential equation of the form T(DV ...,Dm)z=4>{xl,...,xn) admits at least one integral, provided only that satisfies sufficient conditions of continuity, and that the symbolic polynomial T is not identically zero. By suitable labelling, we may ensure that the linear factor of P contains Dv Thus let the equation considered be {(Dx-a2D2-...-amDm-bYQ(Dv ..., Dm)}z = 0, (4)