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15.3 Bessel Functions 539

1 0.5
x
1 2 3 4 5 6 7
0.8 0

0.6 –0.5
0.4 –1

0.2 –1.5
–2
0
0 2 4 6 8 10 12
x –2.5
–0.2
–3
–0.4

FIGURE 15.6 J ν (x) for ν = 0,1,5/3,4. FIGURE 15.7 Y ν (x) for ν = 0,1/3,1/2.

Figures 15.6 and 15.7 show graphs of some Bessel functions of the first and second kinds,
respectively.
Sometimes we encounter disguised versions of Bessel’s equation. The differential equation
2a − 1 a − ν c

2 2 2
2 2 2c−2
y − y + b c x + y = 0 (15.8)
x x 2
has the general solution
c
c
a
a
y(x) = c 1 x J n (bx ) + c 2 x Y n (bx )
if ν = n is an integer, and
a
c
a
c
y(x) = c 1 x J ν (bx ) + c 2 x J ν (bx )
if ν is not an integer. Verification of this is an exercise in chain rule differentiation and is requested
in Problem 1.
EXAMPLE 15.6
We will solve
√
2 3 − 1
61
6
y − y + 784x − y = 0.
x x 2
√
b
Match this equation to the template (15.8). Clearly we need a = 3. Because of the x term, try
2c − 2 = 6, so c = 4. Now we must choose b and ν so that
2
2 2
784 = b c = 16b .
Then b = 2. Next,
2
2 2
2
a − ν c = 3 − 16ν =−61,
so ν = 2. The general solution of this differential equation is
√ 3 4 √ 3 4
y(x) = c 1 x J 2 (7x ) + c 2 x Y 2 (7x ).
Because of the Bessel function of the second kind, this solution is defined for x > 0.

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October 14, 2010 15:20 THM/NEIL Page-539 27410_15_ch15_p505-562