1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

358 CHAPTER. 9 • Z-TR.ANSFOR.MS AND APPLICATI ONS

9.2 Second-Order Homogeneous Difference Equations

Before proceeding with the z-transform method, we mention a heuristic method
based on substitution of a trial solution. Consider the second-order homogeneous
linear constant-coefficient difference equation (HLCCDE)

y[n + 2] - 2ay[n + 1] + by{n] = 0 (9-8)

where a and b are constants. Using the trial solution y(n] = rn, direct substitu-

tion into (9-8) produces the equation rn+^2 - 2arn+I + bT'" = o. Dividing through

by r" produces the characteristic polynomial r^2 - 2ar + b and characteristic

equation

r^2 - 2ar + b = 0. (9-9)

There are three types of solutions to (9-8), which are determined by the nature

of the roots in (9-9).

Case (i) If b < a^2 , then we have real distinct roots r 1 = a - ~ and

r2=a+~.and

y(n] = c1 rf + c2r~. (9-10)

Case (ii) If b = a^2 , then we have a double real root r = r 1 = r 2 =a, and

(9-11)

Case (iii) If b > a^2 , then we have two complex roots r 1 = a - i./b=(i'i and
r2 =a+i~, and

(9-12)

The solution for case (iii) can also be written as the following linear combination:

(9-13)

where r = VfJ and </> = arctan(-.1°;;"-

2

).

Caution. Be sure to use the value of arctan that lies in the range 0 < </> :$ 1r.