Page 560 - Modelling in Transport Phenomena A Conceptual Approach

540 APPENDIX B. SOLUTIONS OF DIFFERENTIAL EQUATIONS

with either k = p - 2 or b = 0, is known as the Bessel’s equation. Solutions to
Bessel’s equations are expressed in the form of power series.

Example B.6 Show that the equation
x2+- y=o
@Y
x-+x-- dy ( :)
2
dx2 dx
is reducible to Bessel’s equation.
Solution
A second-order diflerential equation

can be expressed in the form of Eq. (B.2-16) m follows. Dividing each term in Eq.
(1) by a,(x) gives
@Y
-+--+- al(4 dY a2W = 0
dx2 a,(~) dx a,(x)
The integrating factor, p, is

MzLltiplicntion of Eq. (2) with the integrating factor results in

d(rg)+qY=o (4)
dx
where

To express the given equation in the form of Eq. (B.2-16)’ the first step is to divide
each term by x2 to get

Note that the integrating factor is

Multiplication of Eq. (6) by the integrating factor and rearrangement gives

-
(xs) (x+;x-l)y=o