TABLE OF CONTENTS A linear difference equation of order m with constant coefficients has the form
| (7.10.1) |  |
in which ? 0, a 1,..., ? m along with initial conditions y(0), y(1),..., y( m ? 1) are known constants, and y( m), y( m + 1), y( m + 2)... are unknown. Difference equations are the discrete analogs of differential equations, and, among other ways, they arise by discretizing differential equations. For example, discretizing a second-order linear differential equation results in a system of second-order difference equations as illustrated in Example 1.4.1, p 19. The theory of linear difference equations parallels the theory for linear differential equations, and a technique similar to the one used to solve linear differential equations with constant coefficients produces the solution of (7.10.1) as
| (7.10.2) |  |
in which the ? i 's are the roots of ? m ? ? m ? m -1 - ... ? ? 0 = 0, and the ? i's are constants determined by the initial conditions y(0), y(1),..., y( m ? 1) . The first term on the right-hand side of (7.10.2) is a particular solution of (7.10.1), and the summation term in (7.10.2) is the general solution of the associated homogeneous equation defined by setting ? 0 = 0 .
This section focuses on systems of first-order linear difference equations with constant...
