ABSTRACT
As we have seen through the choice of the equation’s order and homogeneity status, difference equations can be formulated to represent the dependence between the elements of a sequence. This flexibility in the collection of difference equation families can be further enriched by including equations whose coefficients are not constant, but are instead variable. Unfortunately, this generalization may lead to systems of equations for which there are no known general solutions. For example, the work of Chapter 4 reveals that the family of difference equations
yk+2=3yk+1−6yk+k (5.1)
is easily solved, However, the general family of difference equations Pn(k)yk+n+Pn−1(k)yk+n−1+Pn−2(k)yk+n−2+…+P0(k)yk=R(k)
(5.2)
in general cannot be solved when the functions p0(k), P1(k), p2(k)…p3(k) are functions of k. In the types of equations we have solved in Chapter 4, such as the family of third-order equations given by
270yk+4−129yk+3−95yk+2−129yk+1+3yk=11 (5.3)
the coefficients of sequence elements yk to yk+4 are constants with respect to k. This allows a straightforward (although at times complicated) conversion of the infinite sequence of equations represented by equation (5.3) to one equation involving
. Having pointed this out, however, there are several interesting families of difference equations with variable coefficient that are of interest and can be explored with the mathematical tools developed so far in this text.
