4—Differential Equations 93ofx^1 cosxabout the origin? This is singular there, but it’s easy to write the answer anyway because you already
know the series for the cosine.
cosx
x
=
1
x−
x
2+
x^3
24−
x^5
720+···
It starts with the term 1 /xcorresponding tos=− 1 in the Frobenius series.
Assume thata 06 = 0, because that just defines the coefficient of the most negative power,xs. If you allow it
be zero, that’s just the same as redefiningsand it gains nothing. Plug this in to the Bessel differential equation.
x^2 y′′+xy′+ (x^2 −n^2 )y= 0x^2∑∞
k=0ak(k+s)(k+s−1)xk+s−^2 +x∑∞
k=0ak(k+s)xk+s−^1 + (x^2 −n^2 )∑∞
k=0akxk+s= 0∑∞
k=0ak(k+s)(k+s−1)xk+s+∑∞
k=0ak(k+s)xk+s+∑∞
k=0akxk+s+2−n^2∑∞
0akxk+s= 0∑∞
k=0ak[
(k+s)(k+s−1) + (k+s)−n^2]
xk+s+∑∞
k=0akxk+s+2= 0The coefficients of all the like powers ofxmust match, and in order to work out the matches efficiently, and so
as not to get myself confused in a mess of indices, I’ll make an explicit change of the index in the sums.
Let=kin the first sum. Let=k+ 2in the second. Explicitly show the limits of the index on the
sums, or your bound to get it wrong.
∑∞`=0a`[
(`+s)^2 −n^2]
x`+s+∑∞
`=2a`− 2 x`+s= 0The lowest power ofxin this equation comes from the`= 0term in the first sum. That coefficient ofxsmust
vanish. (a 06 = 0)
a 0
[
s^2 −n^2]
= 0
This is called theindicial equation. It determiness, or in this case, maybe twos’s. After this, set to zero the
coefficient ofx+s. a
[
(`+s)^2 −n^2]
+a`− 2 = 0