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MATH 172 Notes Bessel’s Equation
Bessel’s Equation
The family of differential equations known as Bessel equations of order p ≥ 0 look like:
x^2 y′′^ + xy′^ + (x^2 − p^2 )y = 0 , x > 0. (1)
Remarks:
- This is a differential equation (MATH 231). Goal: To find a function y(x) which solves (i.e., satisfies Equation (1). I give this example as an application of power series; it is unlikely to be repeated by a MATH 231 instructor.
- DEs are mathematical models: dv/dt = g, F = m(d^2 s/dt^2 ) (Newton’s 2nd Law), etc. Friedrich Wilhelm Bessel (1784–1846) studied this type of DE in conjunction with considering distur- bances in planetary motion.
- Along with the original context in which they arose, these DEs come up when solving PDEs involving the Laplacian in polar/cylindrical coordinates (such as in understanding the vibrations of a circular drum).
We consider the Bessel equation of order 0:
x^2 y′′^ + xy′^ + x^2 y = 0 , x > 0. (2)
We assume that this equation has a power series solution centered at 0—that is, that the solution y has the form y(x) =
∑^ ∞
n= 0
cnxn,
so that
y′(x) =
∑^ ∞
n= 1
ncnxn−^1 and y′′(x) =
∑^ ∞
n= 2
n(n − 1)cnxn−^2.
Substituting these expressions into Equation (2), we get
0 = x^2 y′′^ + xy′^ + x^2 y = x^2
∑^ ∞
n= 2
n(n − 1)cnxn−^2 + x
∑^ ∞
n= 1
ncnxn−^1 + x^2
∑^ ∞
n= 0
cnxn
∑^ ∞
n= 2
n(n − 1)cnxn^ +
∑^ ∞
n= 1
ncnxn^ +
∑^ ∞
n= 0
cnxn+^2
∑^ ∞
n= 2
n(n − 1)cnxn^ +
c 1 x +
∑^ ∞
n= 2
ncnxn
∑^ ∞
j= 2
cj− 2 xj^ (substituting j = n + 2 in final summation)
= c 1 x +
∑^ ∞
n= 2
[n(n − 1)cn + ncn + cn− 2 ] xn^ = c 1 x +
∑^ ∞
n= 2
(n^2 cn + cn− 2 )xn
= c 1 x + (4c 2 − c 0 )x^2 + (9c 3 + c 1 )x^3 + (16c 4 + c 2 )x^4 + · · · + (n^2 cn + cn− 2 )xn^ + · · ·.
2
MATH 172 Notes Bessel’s Equation
Now, we equate coefficients of various powers of x, noting that the left side of the equation, being zero, has zero x-terms, zero x^2 -terms, zero x^3 -terms, etc.
x^0 : 0 = 0 , x^1 : c 1 = 0 , x^2 : 22 c 2 + c 0 = 0 ⇒ c 2 = −^1 22 c 0 ,
x^3 : 32 c 3 + c 1 = 0 ⇒ c 3 = −^1 32 c 1 = 0 ,
x^4 : 42 c 4 + c 2 = 0 ⇒ c 4 = −^1 22 (2)^2 c 2 =
[
22 (2)^2
] [
c 0
]
(−1)^2
24 (2!)^2
c 0 ,
x^5 : 52 c 5 + c 3 = 0 ⇒ c 5 =
c 3 = 0 ,
x^6 : 62 c 6 + c 4 = 0 ⇒ c 6 =
c 4 =
[ − 1
] [^ (−1) 2
24 (2!)^2
c 0
]
(−1)^3
26 (3!)^2
c 0 , .. . x^2 k−^1 : (2k − 1)^2 c 2 k− 1 + c 2 k− 3 = 0 ⇒ c 2 k− 1 =
(2k − 1)^2 c 2 k− 3 = 0 ,
x^2 k^ : (2k)^2 c 2 k + c 2 k− 2 = 0 ⇒ c 2 k =
22 k^2 c 2 k− 2 = (−1)k 22 k(k!)^2 c 0 ,
and so on. Taking c 0 = K, we have
y(x) = K + 0 x +
22 Kx
(^2) + 0 x (^3) + (−1)^2 24 (2!)^2 Kx^4 + 0 x^5 +
(−1)^3
26 (3!)^2 Kx
= K
[
22 (1!)^2 x
2 + (−1)^2
24 (2!)^2
x^4 +
(−1)^3
26 (3!)^2 x
]
= K ·
∑^ ∞
n= 0
(−1)n 22 n(n!)^2 x
2 n.
The power series
J 0 (x) :=
∑^ ∞
n= 0
(−1)n 22 n(n!)^2 x^2 n,
is called the Bessel function of the first kind of order 0^1.
The method demonstrated above, in which a power series solution is sought for a DE, is commonly used for linear, non-constant coefficient ordinary DEs. A nice exercise (entirely optional) is to adapt this procedure to the solution of Airy’s (differential) equation
y′′^ − xy = 0.
It should give rise to two separate functions^2
y 1 (x) = 1 +
∑^ ∞ n= 1
x^3 n (2 · 3)(5 · 6) · · · [(3n − 1) · (3n)] ,^ and^ y^2 (x)^ =^ x^ +
∑^ ∞ n= 1
x^3 n+^1 (3 · 4)(6 · 7) · · · [(3n) · (3n + 1)].
(^1) There is also a Bessel function of the second kind of order 0. As far as I know, Close Encounters of the Third Kind is still fictional. (^2) See the website http://www.sosmath.com/diffeq/series/series04/series04.html for details.
