2. The Bessel function of order 1 is (-1)*22k 22k+1 k!(k+1)! J1 (x) = x Lk=0 (a) Show that the series converges for all x. (b) Show that the series is a solution of the differential equation x² J" + xJ{ + (x² – 1) J1 = 0. (c) The - Bessel function of order 0 is (-1)*22k Jo (x) = Ek=0 2k (k!)² Show that Jó (x) = –J1 (x).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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2. The Bessel function of order 1 is
(-1)*22k
J1 (x) = x Lk=0 2k+1 k!(k+1)!
(a) Show that
the series converges for all x. (b) Show that the
series is a solution of the differential equation
x² J" + xJ{ + (x²
– 1) Ji = 0. (c) The
-
Bessel function of order 0 is
(-1)*x2k
k=0 2k (k!)?
Jo (x) =
Show that
J¿ (x) = –J1 (x).
Transcribed Image Text:2. The Bessel function of order 1 is (-1)*22k J1 (x) = x Lk=0 2k+1 k!(k+1)! (a) Show that the series converges for all x. (b) Show that the series is a solution of the differential equation x² J" + xJ{ + (x² – 1) Ji = 0. (c) The - Bessel function of order 0 is (-1)*x2k k=0 2k (k!)? Jo (x) = Show that J¿ (x) = –J1 (x).
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