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3.4.1 Example A An easy calculation shows that y = 2*, (2) Yk (1) 1 (3.77) are solutions to the homogeneous equation Yk+2 – 3yk+1 + 2yk = 0. (3.78) Furthermore, their Casoratian is given by the following expression: 2k+1 2k+2 1 1 C(k + 1) = -2k+1. (3.79) Let us use these results to calculate a particular solution to the inhomo- geneous equation Yk+2 – 3yk+1 + 2yk = 4* + 3k². (3.80) Let Y be the particular solution, where Y = C1(k)2* + C2(k). (3.81) The functions C1(k) and C2(k) satisfy the following equations: 2k+1AC1 (k) + AC2(k) = 0, (3.82) 2*+2AC1 (k) + AC2(k) = 4* + 3k2.

3.4.1 Example A An easy calculation shows that y = 2*, (2) Yk (1) 1 (3.77) are solutions to the homogeneous equation Yk+2 – 3yk+1 + 2yk = 0. (3.78) Furthermore, their Casoratian is given by the following expression: 2k+1 2k+2 1 1 C(k + 1) = -2k+1. (3.79) Let us use these results to calculate a particular solution to the inhomo- geneous equation Yk+2 – 3yk+1 + 2yk = 4* + 3k². (3.80) Let Y be the particular solution, where Y = C1(k)2* + C2(k). (3.81) The functions C1(k) and C2(k) satisfy the following equations: 2k+1AC1 (k) + AC2(k) = 0, (3.82) 2*+2AC1 (k) + AC2(k) = 4* + 3k2.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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3.4.1 Example A An easy calculation shows that 9 = 2*, (2) Yk (3.77) are solutions to the homogeneous equation Yk+2 – 3yk+1+2yk = 0. (3.78) Furthermore, their Casoratian is given by the following expression: 2k+1 2k+2 1 1 C(k + 1) = -2k+1. (3.79) Let us use these results to calculate a particular solution to the inhomo- geneous equation Yk+2 3yk+1 + 2yk = 4k + 3k². (3.80) Let Y be the particular solution, where Yg = C1(k)2* + C2(k). (3.81) The functions C1(k) and C2(k) satisfy the following equations: 2k+1AC1 (k) + AC2(k) = 0, (3.82) 2*+2AC1 (k) + AC2(k) = 4* + 3k2.
Transcribed Image Text:3.4.1 Example A An easy calculation shows that 9 = 2*, (2) Yk (3.77) are solutions to the homogeneous equation Yk+2 – 3yk+1+2yk = 0. (3.78) Furthermore, their Casoratian is given by the following expression: 2k+1 2k+2 1 1 C(k + 1) = -2k+1. (3.79) Let us use these results to calculate a particular solution to the inhomo- geneous equation Yk+2 3yk+1 + 2yk = 4k + 3k². (3.80) Let Y be the particular solution, where Yg = C1(k)2* + C2(k). (3.81) The functions C1(k) and C2(k) satisfy the following equations: 2k+1AC1 (k) + AC2(k) = 0, (3.82) 2*+2AC1 (k) + AC2(k) = 4* + 3k2.
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