Solution Of Difference Equations Using Z-transform

Solution of difference equations using Z-transform.

We know that Laplace Transforms are very useful to solve linear differential equations

The Z-transforms are useful to solve linear difference equations.

 Formula:

 

XIV. Problems based on Solution of the difference equations using Z-transform

Solve :

1. Solve y_n+1-2y_n = 0 given y_o = 3

Solution: Given: y_n+1-2y_n = 0

Taking Z-transform on both sides of the difference equation, we get

2. Using Z-transform, solve y_n+2 - 4y_n = 0, given that y₀= 0, y₁ = 2

3. Solve y_n+2-4y_n = 0

4. Using Z-transform, solve

U_n+2 + 3u_n+1 + 2u_n = 0 given u_o = 1, u₁ = 2.

5. Solve the difference equation

y (n+3)- 3y (n + 1) + 2y(n) = 0 given that y(0) = 4, y(1) = 0 and y(2) = 8.

Solution: Given: y (n+3)- 3y (n + 1) + 2y (n) 0

Z [y (n+3)]-3Z [y (n + 1)] + 2Z [y(n)] = 0

[z³ Y(z) - z³y(0) - z² y(1) – zy(2)]

3 [zY(z) - zy(0)] + 2Y(z) 00

[z³ Y(z) — 4z³ - 8z] - 3 [zY(z) — 4z] + 2Y(z) = 0

(z³-3z + 2) Y(z) – 4z3.

[ y (0) = 4, y (1) = 0, y (2) = 8]

6. Solve y_n + 2-2 cos a yn + 1 + yn =0 given that yo  = 1, y1 = cos a.

Solution: Given: y_n + 2-2 cos a yn + 1 + yn =0

Z[y_n+21-2 cos a Z [_{yn +} 1]+ Z[y_n] Z[0]

[z²Y(z) - z²y (0) -zy (1)]-2 cos a [zY (z) - zy (0)] + Y (z) = 0

 

7. Solve the difference equation

y (k + 2) - 4y (k + 1) + 4y (k) = 0 where y(0) = 1, y (1)=0.

Solution :

 

8. Using Z-transform solve y (n) + 3y (n - 1) - 4y (n-2) = 0

n≥ 2 given that y (0) = 3 and y (1) = -2.

Solution : Given : y (n) + 3y (n − 1) −4 y (n − 2) = 0, n ≥ 2

Replace n by n + 2, we get

9. Solve x (n + 1) - 2x (n) = 1, given x(0) = 0

Solution: Given: x (n + 1) - 2x (n) = 1

10. Using Z-transform method solve y_{n + 2} + Y_n = 2 given that yo = y1= 0.

Solution: Given: y_{n + 2} + y_n = 2 and y₀= y₁ = 0

11. Solve y_n+2 + 6y_n+1 + 9y_n = 2ⁿ given y₀= y₁=0

Solution: Given: y_n+2 + 6y_n+1 + 9y_n = 2ⁿ

12. Using the Z-transform, solve

U_n+2 + 4u_n+1 + 3u_n = 2ⁿ with u₀ = u₁ = 1

Solution: Given: U_n+2 + 4u_n+1 + 3u_n = 2ⁿ

13. Using Z-transform, solve u_n+2 − 5u_n+1 + 6u_n = 4ⁿ given_that U₀ = 0, u₁ = 1.

Solution: Given: u_n+2 − 5u_n+1 + 6u_n = 4ⁿ

14. y_{n +2} +4y_{n + 1} + 3y_n =3ⁿ  with y₀= 0, y₁=1)

Solution: Given: y_{n +2} +4y_{n + 1} + 3y_n =3ⁿ 

15. Solve y_n+2 + y_n = n2ⁿ

Solution: Given: y_n+2 + y_n = n2ⁿ

 

16. Using Z-transform, solve y_n+2+4y_n+1-5y_n = 24_n-8 given that y₀ = 3 and y₁ =-5

17. Solve y(n) − y(n − 1) = u(n) + u(n − 1) when_u(n) =n,u(n-1) = n-1/

Solution: Given: y(n) − y(n − 1) = u(n) + u(n − 1)

18. Find the response of the system :

y_{n + 2}-5y_n + ₁ + 6y_n u_n, with y₀ = 0, y₁ and un = n = 0 1, 2,… by Z-transform method.

Solution: Given: y_{n + 2}-5y_n + ₁ + 6y_n =u_n

Taking Z-transform of both sides of the given equation we get

19. Solve the simultaneous difference equations y_n+1 = 5x_n+7;  y_n+1 = x_n+2y_n given that xo = 0, and y₀ = 1.

Solution: Given: (i) x_n+1=5x_n+7

20. Solve the system using Z-transform

x_n+1=7x_n+10y_n

yn₊₁ = x_n+4y_n given that x₀=3, y₀ = 2.

Solution: Given: x_n+1 = 7x_n+10y_n