Essential formulae 671
(ii) differentiate the trial series to find y
and y ,
(iii) substitute the results in the given differential
equation,
(iv) equate coefficients of corresponding powers
of the variable on each side of the equation: this enables index c and coefficients
a 1 , a 2 , a 3 , ... from the trial solution, to be
determined.
Bessel’s equation:
The solution of x 2 d 2 y
dx 2 + x
d y
dx
+ (x 2 − v 2 )y = 0 is:
y = Ax
v
1 −
x
2
2 2 (v + 1)
+
x 4
2 4 × 2! (v + 1)(v + 2)
−
x 6
2 6 × 3! (v + 1)(v + 2)(v + 3)
+ · · ·
+ Bx
−v
1 +
x 2
2 2 (v − 1)
+
x 4
2 4 × 2! (v − 1)(v − 2)
+
x 6
2 6 × 3! (v − 1)(v − 2)(v − 3)
+ · · ·
or, in terms of Bessel functions and gamma functions:
y = AJ v (x) + B J −v (x)
= A
x
2
v
1
(v + 1)
−
x 2
2 2 (1! ))(v + 2)
+
x 4
2 4 (2! ))(v + 4)
− · · ·
+ B
x
2
−v
1
(1 − v)
−
x 2
2 2 (1! ))(2 − v)
+
x
4
2 4 (2! ))(3 − v)
− · · ·
In general terms:
J v (x) =
x
2
v
∞
k=0
(−1) k x 2k
2 2k (k! ))(v + k + 1)
and J −v (x) =
x
2
−v
∞
k=0
(−1) k x 2k
2 2k (k! ))(k − v + 1)
and in particular:
J n (x) =
x
2
n
1
n!
−
1
(n + 1)!
x
2
2
+
1
(2! )(n + 2)!
x
2
4 − · · ·
J 0 (x) = 1 −
x 2
2 2 (1! ) 2 +
x 4
2 4 (2! ) 2
−
x 6
2 6 (3! ) 2 + · · ·
and J 1 (x) =
x
2
−
x 3
2 3 (1! )(2! )
+
x 5
2 5 (2! )(3! )
−
x 7
2 7 (3! )(4! )
+ · · ·
Legendre’s equation:
The solution of (1 − x 2 )
d 2 y
dx 2 − 2x
d y
dx
+ k(k + 1)y = 0
is:
y = a 0
1 −
k(k + 1)
2!
x
2
+
k(k + 1)(k − 2)(k + 3)
4!
x
4
− · · ·
+ a 1
x −
(k − 1)(k + 2)
3!
x
3
+
(k − 1)(k − 3)(k + 2)(k + 4)
5!
x
5
− · · ·
Rodrigue’s formula:
P n (x) =
1
2 n n!
d n (x 2 − 1) n
dx n
Statistics and Probability
Mean, median, mode and standard
deviation:
If x = variate and f = frequency then:
mean ¯
x =
f x
f
The median is the middle term of a ranked set of data.