Pascal Matrices and Particular Solutions to Differential Equations

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. 1 Demonstration of Method. How would one find a particular solution to the following differential equation? y′′′ − y′ + 3y = (1 + 5t)e (1) We begin by defining an operator L so that L = D−D+3. In this instance D is the k derivative of y with respect to t. Note that: L(e) = 63e = p(4)e where p(a) = a − a+ 3. Let us assume a particular solution of y = (c0 + c1t)e . Our strategy will be to differentiate the particular solution and compare it to the right hand side of Equation 1. We can also rewrite the differential equation in matrix format: St. John’s University 1 L(y) = [ 1 5 ] [ 1 t ] e. By applying L to y we obtain the following: L(y∗) = L ( [ c0 c1 ] [ 1 t ]