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...

Extracted 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.