An approach to solving linear constant coefficient difference and differential equations

The underlying structure inherent in the classical method of undetermined coefficients, which is used to obtain particular solutions to linear constant coefficient (LCC) difference or differential equations, is investigated. A system of equations of the form B=M^-1 A is obtained, where B and A are vectors whose elements are the coefficients of the terms in the expressions for the input and solutions, respectively, to the LCC equations. The structure of M that arises for both LCC difference and differential equations, as well as moving-average (FIR) systems, are investigated. It is shown that M is always a lower triangular matrix of order (r-1)×(r+1), where r is the degree of the expressions for the input and solutions. Furthermore, M is characterized by r+1 unique elements, each one defining the diagonal and off-diagonal elements, and is a member of an infinite set of matrices, all of order r+1, which form a group. M can be obtained whenever A and B are given. As a result, if one desires an FIR filter whose output is some linear operation, then the computation of M from A and B imposes a set of necessary and sufficient conditions