Abstract
In this paper any non-homogeneous differential equation with constant
coefficients is reduced to a matrix equation ~q = ~cP. For the discussion, ~q
represents a matrix of constant coefficients to the differential equation, ~c a
matrix of arbitrary constants to the solution, and P is a lower triangular
matrix with entries that are derivatives of the characteristic polynomial
of the differential equation. After careful development, the task becomes
finding a inverse to the matrix P. Interestingly enough, P is a generalized
form of what is termed a Pascal Matrix. [1] An inverse for certain conditions
to such a matrix is proven to exist by the theorem given in the
paper.
This approach was developed in earlier research, [2]. The advantage is
that it uses fundamental concepts such as the linearity of the deriviative,
matrix multiplication, and product rule for derivatives. Futhermore a
precise algorithm to solve a wide variety of differential equations is given
with this approach.