Distribution solutions to ordinary differential equations with polynomial coefficients on the real line

Abstract

In this study we consider the Weyl algebra A1(C) that consists of differential operators P of the form P = pm(x)"m + pm−1(x)"m−1 + . . . + p1(x)" + p0(x), (1) where {p"(x)} are polynomials which in general may have complex coefficients and " is the first order derivation operator with respect to x. To the differential operator P in (1), we associate a differential equation, P(μ) = 0 where μ is either an analytic or a non-analytic function. If the leading polynomial pm(x) has no zeros on the real line (elliptic case) then Cauchy’s result gives m classical solutions. Otherwise, in the non-elliptic case, we have to take into account solutions that are generalized functions (distributions). Our main interest is to establish the dimension of the vector space of the distributional solutions. In this case, we assume that pm(x) has a finite number of real zeros with finite multiplicities and with the degree at most k i.e in the range 1 $ k $ m. With this setting, we will assume that we can re-arrange the corresponding P such that it is locally Fuchsian at each real zero a" of pm(x) withmultiplicitye" ! 1. To each such locally Fuchsian P, we demonstrate the existence of distribution solutions by use of particular cases of P. We state and prove a theorem relating the order of the operator P, the multiplicities of the real zero and the dimension of the P-kernel on the space of distributions Db. Furtherwe prove that for eachreal zero a" of pm(x) there exists a distribution μ supported by the half-line {x ! a"} such that the distribution P(μ) is the Dirac measure at a". Such a μ is called a fundamental solution, and may be used to give a general solution of the differential equation P(μ) = f where f can be either a real analytic function or a generalized function. We also use the theory of boundary values (boundary value distributions) and the Cauchy transform of distributions with compact support to prove coexistence of solutions in the form of the finite order linear combination of the Dirac delta function and its derivatives and rational function solutions to ordinary differential equations with polynomial coefficients.