Page 137 - Advanced engineering mathematics

3.7 Polynomial Coefficients 117

Integrate to obtain
2n + 1
2
2 −(2n+1)/2
ln|W|=− ln(1 + s ) = ln (1 + s ) .
2
Here we have chosen the constant of integration to be zero to obtain a particular solution. Then
2 −(2n+1)/2
W(s) = (1 + s ) .
We must invert W(s) to obtain w(t) and finally y(t). To carry out this inversion, write
−(2n+1)/2

1 1
W(s) = 1 +
s 2n+1 s 2
and use the binomial expansion to obtain
1 2n + 1 1 1 −2n − 1 −2n − 3 1
W(s) = 1 − +
s 2n+1 2 s 2 2! 2 2 s 4
1 −2n − 1 −2n − 3 −2n − 5 1 1
+ + ··· .
4
3! 2 2 2 s s 6
Then
1 2n + 1 1
W(s) = −
s 2n+1 2 s 2n+3
(2n + 1)(2n + 3) 1
+
2(2)(2!) s 2n+5
(2n + 1)(2n + 3)(2n + 5) 1
− + ··· .
2(2)(2)(3!) s 2n+7
Now we can invert this series term by term to obtain
1 2n + 1 t 2(n+1)
2n
w(t) = t −
(2n)! 2 (2(n + 1))!
(2n + 1)(2n + 3) t 2(n+2)
+
2(2)(2!) (2(n + 2))!
(2n + 1)(2n + 3)(2n + 5) t 2(n+3)
− + ··· .
2(2)(2)(3!) (2(n + 3))!
−n
Finally recall that y = t w to obtain the solution
1 2n + 1
−n n n+2
y(t) = t w(t) = t − t
(2n)! 2(2(n + 1))!
(2n + 1)(2n + 3) n+4
+ t
2(2)(2!)((2(n + 2))!)
(2n + 1)(2n + 3)(2n + 5)
− t n+6 + ···
2(2)(2)(3!)(2(n + 3))!
∞ k
(−1) n+2k
= t = J n (t).
2 2k+n k!(n + k)!
k=0
This is the Bessel function of the first kind of order n, usually denoted J n (t) with the choice of
constant made in the integration of the separated variables.
In Section 15.3, we will derive Bessel functions J ν (t) of arbitrary order ν and also second,
linearly independent solutions Y ν (t) to write the general solution of Bessel’s equation of order ν.
We will also develop properties of Bessel functions that are needed for applications.

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October 14, 2010 14:14 THM/NEIL Page-117 27410_03_ch03_p77-120