Explain the determaine 3.4.1 Example AAn easy calculation shows that(1)Yk2k,.(2)= 1Yk(3.77)are solutions to the homogeneous equationYk+2 – 3yk+1 + 2yk = 0.(3.78)Furthermore, their Casoratian is given by the following expression:2k+12k+2 11C(k + 1)1--2*+1(3.79)Let us use these results to calculate a particular solution to the inhomo-geneous equationYk+23yk+1 + 2yk4k + 3k?.(3.80)Let Yk be the particular solution, whereYk = C1 (k)2* + C2(k).(3.81)The functions C1(k) and C2(k) satisfy the following equations:2*+1AC1 (k) + AC2(k) = 0,(3.82)2*+2AC1 (k) + AC2(k) = 4* + 3k².

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Description: Explain the determaine 3.4.1 Example AAn easy calculation shows that(1)Yk2k,.(2)= 1Yk(3.77)are solutions to the homogeneous equationYk+2 – 3yk+1 + 2yk = 0.(3.78)Furthermore, their Casoratian is given by the following expression:2k+12k+2 11C(k + 1)1--

The determinant is given as 2k+112k+21 =2k+1×1-2k+2×1=2k+11-2=-2k+1

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Advanced Math

3.4.1 Example A An easy calculation shows that (1) Yk (3.77) Yk = 1 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) = 2k+1 1 2k+2 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 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, (3.82) 2*+2AC1 (k) + AC2(k) = 4* + 3k?.

3.4.1 Example A An easy calculation shows that (1) Yk (3.77) Yk = 1 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) = 2k+1 1 2k+2 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 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, (3.82) 2+2AC1 (k) + AC2(k) = 4 + 3k?.

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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Question

Explain the determaine

3.4.1 Example A
An easy calculation shows that
(1)
Yk
2k,
.(2)
= 1
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)
1--2*+1
(3.79)
Let us use these results to calculate a particular solution to the inhomo-
geneous equation
Yk+2
3yk+1 + 2yk
4k + 3k?.
(3.80)
Let Yk be the particular solution, where
Yk = C1 (k)2* + C2(k).
(3.81)
The functions C1(k) and C2(k) satisfy the following equations:
2*+1AC1 (k) + AC2(k) = 0,
(3.82)
2*+2AC1 (k) + AC2(k) = 4* + 3k².

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