One solution to the Bessel equation of (nonnegative) integer order N x2 y + xy + (x2

Chapter 1, Problem 52

(choose chapter or problem)

One solution to the Bessel equation of (nonnegative) integer order N x2 y + xy + (x2 N2)y = 0 is y(x) = JN (x) = k=0 (1)k k!(N + k)! x 2 2k+N . (a) Write the first three terms of J0(x). (b) Let J (0, x, m) denote the mth partial sum J (0, x, m) = m k=0 (1)k (k!)2 x 2 2k . Plot J (0, x, 4) and use your plot to approximate the first positive zero of J0(x). Compare your value against a tabulated value or one generated by a computer algebra system. (c) Plot J0(x) and J (0, x, 4) on the same axes over the interval [0, 2]. How well do they compare? (d) If your system has built-in Bessel functions, plot J0(x) and J (0, x, m) on the same axes over the interval [0, 10] for various values of m. What is the smallest value of m that gives an accurate approximation to the first three positive zeros of J0(x)?