Bessel’s equation isx2y′′ + xy′ + (x2 − n2)y = 0,where n is a scalar, y = y(x) is a function of the independent variable x, and y′ and y′′ are the first and second derivatives of y with respect to x.a) Show that x0 = 0 is a regular singular point. b) What does Fuch’s theorem have to say about a series solution about x0= 0? c) Consider the ansatz y(x) = xk Σ∞m=0 amxm, with a0 =/= 0, and k and the coefficients {am}are to be determined. Show that this ansatz, when used in Bessel’s equation, leads to therecurrence relation am[(m + k)2 - n2] = -am-2 (part c shown in attachment) c) Consider the ansatz y(x) = xo amam, with ao # 0, and k and the coefficients {m}are to be determined. Show that this ansatz, when used in Bessel's equation, leads to therecurrence relationam[(m + k)² = n²] ==-am-2-

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Answered: Bessel’s equation is x2y′′ + xy′ + (x2…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

Bessel’s equation is

x²y′′ + xy′ + (x² − n²)y = 0,

where n is a scalar, y = y(x) is a function of the independent variable x, and y′ and y′′ are the first and second derivatives of y with respect to x.
a) Show that x₀ = 0 is a regular singular point.

b) What does Fuch’s theorem have to say about a series solution about x₀= 0?

c) Consider the ansatz y(x) = x^k Σ^∞_m=0 a_mx^m, with a₀ =/= 0, and k and the coefficients {a_m}
are to be determined. Show that this ansatz, when used in Bessel’s equation, leads to the
recurrence relation a_m[(m + k)²- n²] = -a_m-2(part c shown in attachment)