Consider the problem ( xy_)_ + k2 x y = xy, y and y_ bounded as x 0+, y(1) = 0, where k

Chapter 11, Problem 3

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Consider the problem ( xy_)_ + k2 x y = xy, y and y_ bounded 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 to Bessels equation of order k (see of Section 5.7). One solution is Jk (t); a second linearly independent solution, denoted by Yk (t), is unbounded as t 0. b. Show formally that the eigenvalues 1, 2, . . . of the given problem are the squares of the positive zeros of Jk_ _ and that the corresponding eigenfunctions are n( 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 relation _ 1 0 xm( x)n( x)dx = 0, m _= n. d. Determine the coefficients in the formal series expansion f ( x) = _ n=1 ann( x). e. Find a formal solution of the nonhomogeneous problem ( xy_)_ + k2 x y = xy + f ( x), y and y_ bounded as x 0+, y(1) = 0, where f is a given continuous function on 0 x 1, and is not an eigenvalue of the corresponding homogeneous problem.