Consider the Bessel equation of order x2y + 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 x2y + 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 equationare and .(b) Corresponding to the larger root , show that one solution isy1(x) = x1 11!(1 + )x22+12!(1 + )(2 + )x24+ m=3(1)mm!(1 + )(m + )x22m.(c) If 2 is not an integer, show that a second solution isy2(x) = x1 11!(1 )x22+12!(1 )(2 )x24+ m=3(1)mm!(1 )(m )x22m.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 notan integer