ECE 351 – Linear Systems II Homework #2 Solve the following linear difference equations for yk, k≥0: 1. yk−58yk −1+332yk − 2= 0; y0= 1, y1= 0 [ Ans: ⇒ yk= −238#$%&'(k+ 314#$%&'(k, k ≥ 0 ] 2. yk+12yk −1= −12"#$%&'k; y0= 2 [ Ans: yk= 2 −12"#$%&'k+ k −12"#$%&'k] 3. yk+12yk −1= k −12"#$%&'k; y0= 2 [ Ans: yk= 2 −12"#$%&'k+12k −12"#$%&'k+12k2−12"#$%&'k] 4. yk+ yk −1+14yk −2= −12"#$%&'k; y0= −1, y1= 0 [ Ans: yk= −1 −12"#$%&'k+12k −12"#$%&'k+12k2−12"#$%&'k] 5. yk+ yk −1+14yk −2= k −12"#$%&'k; y0= 1, y1= 0[ Ans: yk=1 −12"#$%&'k−53k −12"#$%&'k+12k2−12"#$%&'k+16k3−12"#$%&'k] 6. yk − 2− 6yk −1+ 8yk= 9; y0= 4, y1=154 [ Ans: yk= 212!"#$%&k−14!"#$%&k+ 3, k ≥ 0] 7. yk + 2+ 4yk= 16 2( )k; y0= y1= 1 [ Ans: yk= 2( )k−cosπ2k −32sinπ2k#$%&'(+ 2 2( )k, k ≥ 0] 8. yk + 2− 2yk +1+ 2yk= 0; y0= 2, y2= 2 [ Ans: yk= 2( )k2 cosπ4k + sinπ4k"#$%&', k ≥ 0] 9. yk +1+22yk= cosπ4kuk; y0= 0 [ Ans: yk= −45 2−22"#$%&'k+45 2cosπ4k +25 2sinπ4k, k ≥ 0] 10. yk +1+ 6 yk= 12uk+ 2uk −1 [ Ans: yk= −53−6( )k+ 2"#$%&'uk−1] 11. yk + 2− 2 yk +1+ yk= 6 + 12k( )uk [ Ans: yk= k − 3k2+ 2k3"#$%uk]12. yk + 2+ 3yk +1+ 2yk= 4 sinπ2k + sinπ2(k + 1), y0=58, y1= −18 [ Ans: yk= −2 −2( )k+ 4 −1( )k−118cosπ2k + −18sinπ2k] Before attempting problems 13 - 16, read and try the examples in MATLAB Tutorial #2. For these three problems turn in your MATLAB statements and the plots. 13. Solve the difference equation in problem 4 recursively for 0 ≤ k ≤ 10. Plot the recursive solution found here with the closed form solution found in problem 4 right below it (using subplot). 14. For the difference equation in problem 9: a. Find the solution recursively for 0 ≤ k ≤ 20. b. Find the solution for 0 ≤ k ≤ 20 using the filter command. Note that the difference equation must be rewritten so that the largest subscript is “k” before the vectors a and b are formed. Replacing k with k-1 in each term can do this. c. Plot the solutions found in part a and b along with the solution obtained in problem 9 and compare. Use the subplot command. 15. Solve the difference equation in problem 10 using the filter command for 0≤k≤4. Plot the solution found here above the closed form solution found in problem 10 (using subplot). 16. Solve the difference equation in problem 11 using the filter command for 0 ≤ k ≤ 5. Plot the solution found here above the closed form solution found in problem 11 (using