082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

592 Chapter 19 Ordinary differential equations I
EXERCISES19.8.2
1 Write outexplicitly the firstfiveterms in the series
expansion ofJ 0 (x) andthe firstfour terms in the
expansion ofJ 1 (x).BydifferentiatingJ 0 (x) verify
that d dx (J 0 (x)) = −J 1 (x).
2 (Method ofFrobenius) Substituteapower series in
∞∑
the more general formy = a m x m+r ,whereris to
m=0
be determined, intoBessel’sequationoforder 1:
x 2d2 y
dx 2 +xdy dx +(x2 −1)y=0.
(i) By equatingcoefficients ofx r show thatr
satisfies theindicialequationr 2 −1 = 0 and
hence deducer = ±1.
(ii) Continuethe solution forthe caser = 1 to
obtain a seriessolution ofthe differential
equation. Findthe first 3 non-zeroterms.
(iii) Confirm that the solution is equivalent to that
obtained previously in Equation19.29.
19.8.3 Bessel’sequationandBesselfunctionsofhigherorder
Tocopewithmoregeneralcases,includingthoseofanon-integerorder,theusualpractice
is to seek solutions in the form a m
∞∑
x m+r where r is an unknown real number
m=0
that is chosen during the course of the calculation in order to generate a solution (see
Exercise 19.8.2, question 2). For details on how to proceed in such cases you should
consultanadvanced texton differential equations.
When the order, v, is a positive integer, it is conventional to label it asninstead. A
solution of Bessel’s equation of positive integer ordernis a Bessel function of the first
kind ofordern, denotedJ n
(x), and can beshown tobe
J n
(x) =
∞∑
m=0
(−1) m x 2m+n
2 2m+n m!(m +n)! , forn=0,1,2,...
Graphs of these functions forn = 0,1,2,3,4 are shown inFigure 19.18.
Whilst to this point we have only discussed cases where the order is non-negative,
in fact Bessel functions are defined for any order v, −∞ < v < ∞. Later, in Engineering
application 19.13 we shall make use of Bessel functions with negative integer
order. In such cases, careful inspection of the above definition shows that it will fail.
When n is negative, then m + n will be negative for some values of m and in these
case (m +n)! is not defined. For a complete treatment, and one which is beyond the
scope of this book, it is necessary to define a new function - the gamma function Ŵ(x)
- which can be thought of as a generalisation of factorials to deal with negative arguments.
Nevertheless, this is possible and it can be shown that Bessel functions with
negativeintegerorderarerelatedtotheircounterpartswithpositiveintegerorderbythe
result
J −n
(x) = (−1) n J n
(x).
(See Exercise 19.8.3)