Consider the boundary value problem ( xy_)_ = xy, y and y_ bounded as x 0+, y_(1) = 0

Chapter 11, Problem 2

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Consider the boundary value problem ( xy_)_ = xy, y and y_ bounded as x 0+, y_(1) = 0. a. Show that 0 = 0 is an eigenvalue of this problem corresponding to the eigenfunction 0( x) = 1. If > 0, show formally that the eigenfunctions are given by the Bessel functions n( x) = J0_ n x_, where n is the nth positive root (in increasing order) of the equation J _ 0_ _ = 0. It is possible to show that there is an infinite sequence of such roots. G b. Create a graph supporting the claim that J _ 0_ _ = 0 has an infinite sequence of positive roots. c. Show that if m, n = 0, 1, 2, . . . , then _ 1 0 xm( x)n( x)dx = 0, m _= n. d. Find a formal solution to the nonhomogeneous problem ( xy_)_ = 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.