Difference Equations Example E Consider the third-order difference equation Yk+3 – 7yk+2 + 16yk+1 – 12yk = k2*. (4.174) Its characteristic equation is p3 – 7r2 + 16r – 12 = (r – 2) (r – 3) = 0, (4.175) and the corresponding homogeneous solution is (H) (cı + c2k)2* + c33k, (4.176) || k2k is where c1, c2, and c3 are arbitrary constants. The family of Rk [2*, k2*]. Note that both members occur in the homogeneous solution; there- fore, we must multiply the family by k2 to obtain a new family that does not contain any function that appears in the homogeneous solution. The new family is [k2*, k³2*]. Thus, the particular solution is || (P) = (Ak² + Bk³)2*, (4.177) where A and B are to be determined. The substitution of equation (4.177) into equation (4.174) gives 2*(-8A+24B) + k2*(-24B) = k2*, (4.178) or 8A – 24B = 0, -24B = 1, (4.179) and A = -1/8, B = -1/24. (4.180) Therefore, the particular solution is (P) Yk -1/24(3 + k)k²2k, (4.181) and the general solution is Yk = (c1 + c2k)2* + c33k – 1/24(3 + k)k²2*. (4.182)
Difference Equations Example E Consider the third-order difference equation Yk+3 – 7yk+2 + 16yk+1 – 12yk = k2*. (4.174) Its characteristic equation is p3 – 7r2 + 16r – 12 = (r – 2) (r – 3) = 0, (4.175) and the corresponding homogeneous solution is (H) (cı + c2k)2* + c33k, (4.176) || k2k is where c1, c2, and c3 are arbitrary constants. The family of Rk [2*, k2*]. Note that both members occur in the homogeneous solution; there- fore, we must multiply the family by k2 to obtain a new family that does not contain any function that appears in the homogeneous solution. The new family is [k2*, k³2*]. Thus, the particular solution is || (P) = (Ak² + Bk³)2*, (4.177) where A and B are to be determined. The substitution of equation (4.177) into equation (4.174) gives 2*(-8A+24B) + k2*(-24B) = k2*, (4.178) or 8A – 24B = 0, -24B = 1, (4.179) and A = -1/8, B = -1/24. (4.180) Therefore, the particular solution is (P) Yk -1/24(3 + k)k²2k, (4.181) and the general solution is Yk = (c1 + c2k)2* + c33k – 1/24(3 + k)k²2*. (4.182)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Topic Video
Question
Explain the determine
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,