Page 562 - Modelling in Transport Phenomena A Conceptual Approach

542 APPENDIX B. SOLUTIONS OF DIFFERENTIAC EQUATIONS

Case (ii) a < 0

In this case the solution is given by
y = xaB [CIIn(Qx") + C21-n(Rxa)] n # integer (B.2-25)

y = xa@ [CIIn(Rxa) + C2Kn(Rxa)] n = integer (B.2-26)
where C1 and C2 are constants, and R is defined by
Q=-a- .A (B.2-27)
a
The term In(x) is known as the modified Bessel function of the first kind of order
-
n and is given bv
O0 (x/2)2i+n
In(x) = i! r(i + n + 1) (B.2-28)
i=O
I-n(x) is obtained by simply replacing n in Eq. (B.2-28) with -n. When n is
not an integer, the functions In(x) and I-n(x) are linearly independent solutions
of Bessel's equation as given by Q. (B.2-25). However, when n is an integer, the
functions In(x) and I+(X) are linearly dependent. In this case, the solution is
given by Eq. (B.2-26) in which Kn(x) is known as the modified Bessel finction of
the second kind of order n and is given by

(B.2-29)

Example B.7 Obtain the general solution of the following equations in terns of
Bessel functions:
dY
a)x--3-+zy=O
8Y
dx2 dx
8Y
b) dz2 - X'Y = 0
Solution
a) Note that the integrating factor is x-~ and the equation can be rewritten as
d
dx
Therefore, p = - 3; a = 1; j = - 3; b = 0. Since b = 0, the equation is reducible
to Bessel's equation. The terms a, /3, and n are calculated from Eqs. (B.2-17)-
(B.2-19) as
2-p+j
a=
2
-
- 2+3-3 =I
2