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Question 3 (a) Show that the equation and boundary conditions d 22 dy da dx + Ary = 0, y(1) = 0, y'(2) = 0, form a regular Sturm-Liouville system. Show further that the system can be written as a constrained variational problem with functional S[y) = | dx x?y?, y(1) = 0, and constraint C[y) = dr ry = 1. (b) Assume that the eigenvalues k and eigenfunctions yk, k = 1, 2,..., exist. Working from equation (1), derive the relationship k = 1,2, .... (c) Using the trial function z = Asin(7(x – 1)/2), show that the smallest eigenvalue, A1, satisfies the inequality (7m2- 18)교2 6(4 + 372) Justify your answer briefly.

Question 3 (a) Show that the equation and boundary conditions d 22 dy da dx + Ary = 0, y(1) = 0, y'(2) = 0, form a regular Sturm-Liouville system. Show further that the system can be written as a constrained variational problem with functional S[y) = | dx x?y?, y(1) = 0, and constraint C[y) = dr ry = 1. (b) Assume that the eigenvalues k and eigenfunctions yk, k = 1, 2,..., exist. Working from equation (1), derive the relationship k = 1,2, .... (c) Using the trial function z = Asin(7(x – 1)/2), show that the smallest eigenvalue, A1, satisfies the inequality (7m2- 18)교2 6(4 + 372) Justify your answer briefly.

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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Question 3 (a) Show that the equation and boundary conditions d dy + Axy = 0, y(1) = 0, y'(2) = 0, dx dr form a regular Sturm-Liouville system. Show further that the system can be written as a constrained variational problem with functional S\y] = dr 2*y%, y(1) = 0, and constraint -2 C[y] = / dr zy² = 1. (b) Assume that the eigenvalues k and eigenfunctions yk, k = 1, 2, ..., exist. Working from equation (1), derive the relationship dk = [ dx a*yf, k= 1,2,... 2.12 k = 1, 2, .... (c) Using the trial function z = A sin(7(x – 1)/2), show that the smallest eigenvalue, A1, satisfies the inequality (7л? — 18)т? 6(4 + 3т?) Justify your answer briefly.
Transcribed Image Text:Question 3 (a) Show that the equation and boundary conditions d dy + Axy = 0, y(1) = 0, y'(2) = 0, dx dr form a regular Sturm-Liouville system. Show further that the system can be written as a constrained variational problem with functional S\y] = dr 2*y%, y(1) = 0, and constraint -2 C[y] = / dr zy² = 1. (b) Assume that the eigenvalues k and eigenfunctions yk, k = 1, 2, ..., exist. Working from equation (1), derive the relationship dk = [ dx a*yf, k= 1,2,... 2.12 k = 1, 2, .... (c) Using the trial function z = A sin(7(x – 1)/2), show that the smallest eigenvalue, A1, satisfies the inequality (7л? — 18)т? 6(4 + 3т?) Justify your answer briefly.
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