Using Z-transform, solve the difference equation: u_(n+2)-8u_(n+1)+12u_n=1 with u_0=0, u_1=2

Ibrahim Rosales

Ibrahim Rosales

Answered question

2022-07-16

Using Z-transform, solve the difference equation:
u n + 2 8 u n + 1 + 12 u n = 1 , with u 0 = 0 , u 1 = 2.

Answer & Explanation

Dominique Ferrell

Dominique Ferrell

Beginner2022-07-17Added 18 answers

Step 1
Given that u n + 2 8 u n + 1 + 12 u n = 1, with u 0 = 0 , u 1 = 2
Now let's find the root,
m 2 8 m 12 = 0
m 2 6 m 2 m + 12 = 0
m = 6 , 2
So u n = A × 2 n + B × 6 n
For the non-homogenous part,
u n = a
Then,
a 8 a + 12 a = 1
5 a = 1
a = 1 5
Step 2
Given that u ( 0 ) = 0 ,
0 = A + B + 1 5
A + B = 1 5 = 2
And u 1 = 2
2 = 2 A + 6 B + 1 5
2 A + 6 B = 9 5
By multiplying 2 with (1) and subtranct (1)-(2)
6 B 2 B = 9 5 + 2 5
4 B = 11 5
B = 11 2
B = 0.55
So
A + 0.55 = 0.2
A = 0.75
So the z-transform for the
u n + 2 8 u n + 1 + 12 u n = 1 is,
u n = 0.75 × 2 n + 0.55 × 6 n + 1 5

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