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) = 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 the recurrence relation am[(m + k)2 - n2] = -am-2   (part c shown in attachment)

Advanced Engineering Mathematics
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ISBN:9780470458365
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Chapter2: Second-order Linear Odes
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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 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 Σm=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 the
recurrence 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 the
recurrence relation
am[(m + k)² = n²] =
=-am-2-
Transcribed Image Text: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 the recurrence relation am[(m + k)² = n²] = =-am-2-
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