On a correlation between differential equations and their characteristic equations

Abstract

Many theoretical and practical problems from theory and practice acquire resolving trough plenty of concrete process characteristics. Problems of this type are eigenvalues problems, boundary value problems, optimal values at variation calculation, of polynomials existence as special functions with determined characteristic. Classical theory of partial differential equations from mathematical physics is related with the functions of Lame, Mathieu, classical orthogonal polynomials, polynomials of Appell, polynomials of Stieltjes, etc. Heine problem for the number of linear differential equations with polynomial coefficients that have polynomial solution, connected with the practical problem of equilibrium (Stieltjes [10]), is known [6,7]. The connection between roots of characteristic algebraic equation with the form of solutions of homogenous linear differential equation with constant coefficients, is already known in classical theory of differential equations. The similar result for existence of polynomial solution of linear homogenous differential equation with polynomial coefficients is obtained [8].