1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

9.2 • SECOND-OR.DER. HOMOGENEOUS DIFFER.ENCE EQUATIONS 369

Solution

(a) The homogeneous difference equation has the form (9-8) with a = :£f

and b = ; and b > a^2 so that the solutions are complex and have the

form

where

(4 2

r = Vb = y 9 = 3 and

(

<f> = arctan ~) a

=Mot'° ( :,~ ~ - ( -7)')

7T

= arctan(l) =

4

.

Hence, the general homogeneous solution is

(

y,,[n] = c1 2)n 7T (2)n 1f

3

cos(

4

n) + c2

3

sin(

4

n),

and is illustrated in Figure 9.2.

Remark 9. 12
The homogeneous solution is transient and will decay to 0 as n ....., oo; i.e.,

limn->oo Yh (n] = 0. •

y[11]

JO

8

6

4

2

• 2
  -4

Figure 9.2 A typical solution to y[n + 2] - ~ J2y[n + 1) + h ln) = O.