Linear Difference Equations with Constant Coefficients
Linear difference equations with constant coefficients can be solved in much the same way as linear differential equations with constant coefficients. The characteristics of the two types of solutions are similar but not identical.
Consider the nth-order homogeneous finite difference equation with constant coefficients:
yk+n + a1yk+n-1 + a2yk+n-2 + + anyk = 0, (1.3)
where ax, a2,…, an are constants. The general solution of such an equation has the form:
Ук = crk, (1.4)
where c and r are constants. Substitution of Eq. (1.4) into Eq. (1.3) yields, after factoring out the common factor crk,
f (r) = rn + a1rn-1 + a2rn-2 +—————– + an-1r + an = 0. (1.5)
Here, f(r) is an nth-order polynomial and thus has n roots ri, i = 1,2,…, n. For each ri we have a solution:
yk = cirk> (1.6)
where ci is an arbitrary constant. The most general solution may be found by superposition.