3.4.1 Example A An easy calculation shows that y0 = 2*, Yk (1) %3D (2) = 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 1 2k+2 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 Yk = C1(k)2* + C2(k). (3.81) The functions C1(k) and C2(k) satisfy the following equations: 2k+1AC1(k) + AC2(k) = 0, %3D (3.82) 2*+2AC1 (k) + AC2(k) = 4* + 3k2.
3.4.1 Example A An easy calculation shows that y0 = 2*, Yk (1) %3D (2) = 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 1 2k+2 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 Yk = C1(k)2* + C2(k). (3.81) The functions C1(k) and C2(k) satisfy the following equations: 2k+1AC1(k) + AC2(k) = 0, %3D (3.82) 2*+2AC1 (k) + AC2(k) = 4* + 3k2.
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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Explain thedetermaine
Transcribed Image Text:3.4.1 Example A
An easy calculation shows that
Yk
2*, Y
(2)
= 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:
C(k + 1) =+2 =-2*+1.
2k+1 1
2k+2 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 Yk 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.
-
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