3.2 General Results for Linear Equations The linear equation of the nth order is Pn(t)y(t + n) + + po(t)y(t) = r(t), (3.4) where po(t),..., Pn (t) and r(t) are assumed to be known and po(t) = 0, Pn(t) 0 for all t. If r(t) # 0, we say that Eq. (3.4) is "nonhomogeneous." As in Section 3.1, we will study Eq. (3.4) in association with the corresponding homogeneous equation Pn(t)u(t + n) + + po(t)u(t) = 0. (3.4') Note that Eq. (3.4) can also be written using the shift operator as (Pn(1) E" + + Po(t)E) y(t) = r(t), where EO 1. Since E = A + 1, it is also possible to write Eq. (3.4) in terms of the difference operator. However, the following example shows that the order of the equation is not apparent in that case. Theorem 3.8. Let ui(t),.. un (t) be independent solutions of Eq. (3.4'). Then y(t) = a(t)u(t) + + an(t)un(t) is a solution of Eq. (3.4), provided a,, an satisfy the matrix equation Aai(t)" W(t + 1) Aan(t) 0 3.45 Use Theorem 3.8 to solve the first order equation y(t+1) - 2y(t) (3). 2