Consider the boundary value problem(xy)= xy,y, ybounded as

Chapter 11, Problem 2

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Consider the boundary value problem(xy)= xy,y, ybounded as x 0, y(1) = 0.(a) Show that 0 = 0 is an eigenvalue of this problem corresponding to the eigenfunction0(x) = 1. If > 0, show formally that the eigenfunctions are given by n(x) = J0(n x),where n is the nth positive root (in increasing order) of the equation J0() = 0. It ispossible to show that there is an infinite sequence of such roots.(b) Show that if m, n = 0, 1, 2, ... , then10xm(x)n(x) dx = 0, m = n.(c) Find a formal solution to the nonhomogeneous problem(xy)= 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.