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 o=0 is a regular singular point. =0? b) What does Fuch's theorem have to say about a series solution about To = c) Consider the ansatz y(x) = xm-0 amxm, with ao # 0, and k and the coefficients {am} are to be determined. Show that this ansatz, when used in Bessel's equation, leads to the recurrence relation am[(m + k)² -n²] = -am-2.
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 o=0 is a regular singular point. =0? b) What does Fuch's theorem have to say about a series solution about To = c) Consider the ansatz y(x) = xm-0 amxm, with ao # 0, and k and the coefficients {am} are to be determined. Show that this ansatz, when used in Bessel's equation, leads to the recurrence relation am[(m + k)² -n²] = -am-2.
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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![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 o=0 is a regular singular point.
=0?
b) What does Fuch's theorem have to say about a series solution about To =
c) Consider the ansatz y(x) = xm-0 amxm, with ao # 0, and k and the coefficients {am}
are to be determined. Show that this ansatz, when used in Bessel's equation, leads to the
recurrence relation
am[(m + k)² -n²] = -am-2.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffade91a1-6997-42f9-8725-9969e19df335%2F67fdc103-7e1a-433f-bab7-7d607455dc6d%2F3hdtnz_processed.png&w=3840&q=75)
Transcribed Image Text: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 o=0 is a regular singular point.
=0?
b) What does Fuch's theorem have to say about a series solution about To =
c) Consider the ansatz y(x) = xm-0 amxm, with ao # 0, and k and the coefficients {am}
are to be determined. Show that this ansatz, when used in Bessel's equation, leads to the
recurrence relation
am[(m + k)² -n²] = -am-2.
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