dx de (22 dy) + λxy = 0, y(1) = 0, y′(2) = 0, form a regular Sturm-Liouville system. further that the system can be written as a constrained variational problem with functional S[y] = 2 {{² [² dx x²y'², y(1) = 0, and constraint 2 C{[y] = [{² da xy² = 1. dx Assume that the eigenvalues λ and eigenfunctions yk, k = 1, 2, . . ., exist. Working from equation (1), derive the relationship λι = 2 ♫² da ✓ dx x²y, k = 1, 2, ………. - Using the trial function z = A sin(л(x − 1)/2), show that the smallest eigenvalue, A1, satisfies the inequality (7π² - 18)π² λι Σ 6(4+32) Justify your answer briefly.
dx de (22 dy) + λxy = 0, y(1) = 0, y′(2) = 0, form a regular Sturm-Liouville system. further that the system can be written as a constrained variational problem with functional S[y] = 2 {{² [² dx x²y'², y(1) = 0, and constraint 2 C{[y] = [{² da xy² = 1. dx Assume that the eigenvalues λ and eigenfunctions yk, k = 1, 2, . . ., exist. Working from equation (1), derive the relationship λι = 2 ♫² da ✓ dx x²y, k = 1, 2, ………. - Using the trial function z = A sin(л(x − 1)/2), show that the smallest eigenvalue, A1, satisfies the inequality (7π² - 18)π² λι Σ 6(4+32) Justify your answer briefly.
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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![dx
de (22 dy)
+ λxy = 0, y(1) = 0, y′(2) = 0,
form a regular Sturm-Liouville system.
further that the system can be written as a constrained
variational problem with functional
S[y] =
2
{{²
[²
dx x²y'², y(1) = 0,
and constraint
2
C{[y] = [{² da xy² = 1.
dx
Assume that the eigenvalues λ and eigenfunctions yk, k = 1, 2, . . .,
exist. Working from equation (1), derive the relationship
λι
=
2
♫² da
✓
dx x²y, k = 1, 2, ……….
-
Using the trial function z = A sin(л(x − 1)/2), show that the smallest
eigenvalue, A1, satisfies the inequality
(7π² - 18)π²
λι Σ
6(4+32)
Justify your answer briefly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa6c8ed7d-75cc-4e27-869e-3ad6a1efc0b4%2F8d35acee-0cfa-4154-b43c-465db4f421e9%2F12wxxye_processed.jpeg&w=3840&q=75)
Transcribed Image Text:dx
de (22 dy)
+ λxy = 0, y(1) = 0, y′(2) = 0,
form a regular Sturm-Liouville system.
further that the system can be written as a constrained
variational problem with functional
S[y] =
2
{{²
[²
dx x²y'², y(1) = 0,
and constraint
2
C{[y] = [{² da xy² = 1.
dx
Assume that the eigenvalues λ and eigenfunctions yk, k = 1, 2, . . .,
exist. Working from equation (1), derive the relationship
λι
=
2
♫² da
✓
dx x²y, k = 1, 2, ……….
-
Using the trial function z = A sin(л(x − 1)/2), show that the smallest
eigenvalue, A1, satisfies the inequality
(7π² - 18)π²
λι Σ
6(4+32)
Justify your answer briefly.
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