Consider the Bessel equation of order x2 y + xy + (x2 2 )y = 0, x > 0, where is real and

Chapter 5, Problem 9

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Consider the Bessel equation of order x2 y + xy + (x2 2 )y = 0, x > 0, where is real and positive. (a) Show that x = 0 is a regular singular point and that the roots of the indicial equation are and . (b) Corresponding to the larger root , show that one solution is y1(x) = x 1 1 1!(1 + ) x 2 2 + 1 2!(1 + )(2 + ) x 2 4 + m=3 (1)m m!(1 + )(m + ) x 2 2m . (c) If 2 is not an integer, show that a second solution is y2(x) = x 1 1 1!(1 ) x 2 2 + 1 2!(1 )(2 ) x 2 4 + m=3 (1)m m!(1 )(m ) x 2 2m . Note that y1(x) 0 as x 0, and that y2(x) is unbounded as x 0. (d) Verify by direct methods that the power series in the expressions for y1(x) and y2(x) converge absolutely for all x. Also verify that y2 is a solution provided only that is not an integer.