Systems of Difference Equations with Constant Coefficients

Abstract

The reader will recall that the second-order difference equation

$${{a}_{0}}{{y}_{t}} + {{a}_{1}}{{y}_{{t + 1}}} + {{a}_{2}}{{y}_{{t + 2}}} = g(t + 2),$$

((78))

where y _t is the scalar dependent variable, the a _h i = 0,1,2, are the (constant) coefficients, and g(t) is the real-valued “forcing function,” is soived in two steps. First we consider the homogeneous part

$${{a}_{0}}{{y}_{t}} + {{a}_{1}}{{y}_{{t + 1}}} + {{a}_{2}}{{y}_{{t + 2}}} = 0,$$

and find the most general form of its solution, called the general solution to the homogeneous part. Then we find just one solution to the equation in (78), called the particular solution. The sum of the general solution to the homogeneous part and the particular solution is said to be the general solution to the equation. What is meant by the “general solution,” denoted, say, by y_f*, is that y* satisfies (78) and that it can be made to satisfy any prespecified set of “initial conditions.” To appreciate this aspect rewrite (78) as

$${{y}_{{t + 2}}} = \bar{g}(t + 2) + {{\bar{a}}_{1}}{{y}_{{1 + 1}}} + {{\bar{a}}_{0}}{{y}_{t}},$$

((79))

where, assuming a ₂ ≠ 0,

$$\bar{g}\left( {t + 2} \right) = \frac{1}{{{{a}_{2}}}}g\left( {t + 2} \right),\quad {{\bar{a}}_{1}} = \frac{{{{a}_{1}}}}{{{{a}_{2}}}},\quad {{\bar{a}}_{0}} = \frac{{{{a}_{0}}}}{{{{a}_{2}}}}.$$