Consider the problem(xy)+ (k2/x)y = xy,y, ybounded as x 0,

Chapter 11, Problem 3

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Consider the problem(xy)+ (k2/x)y = xy,y, ybounded as x 0, y(1) = 0,where k is a positive integer.(a) Using the substitution t = x, show that the given differential equation reduces toBessels equation of order k (see of Section 5.7). One solution is Jk(t); a secondlinearly independent solution, denoted by Yk(t), is unbounded as t 0.(b) Show formally that the eigenvalues 1, 2, ... of the given problem are thesquares of the positive zeros of Jk() and that the corresponding eigenfunctions aren(x) = Jk(n x). It is possible to show that there is an infinite sequence of such zeros.(c) Show that the eigenfunctions n(x) satisfy the orthogonality relation10xm(x)n(x) dx = 0, m = n.(d) Determine the coefficients in the formal series expansionf(x) = n=1ann(x).(e) Find a formal solution of the nonhomogeneous problem(xy)+ (k2/x)y = xy + f(x),y, ybounded as x 0, y(1) = 0,where f is a given continuous function on 0 x 1, and is not an eigenvalue of thecorresponding homogeneous problem.