Mathematical Methods for Physicists: A concise introduction - Site Map

SOLUTIONS IN POWER SERIES
Hence the last equation can be written
u 00 ‡ 2r ‡ g
0
‡ u 0 ˆ 0:
x
…2:32†
Since r ˆ…1 g 0 †=2 the term …2r ‡ g 0 †=x equals 1=x, and by dividing by u 0 we
thus have
By integration we obtain
u 00
u 0 ˆ 1 x ‡:
ln u 0 ˆln x ‡ or u 0 ˆ 1
x e…...† :
Expanding the exponential function in powers of x and integrating once more, we
see that the expression for u will be of the form
u ˆ ln x ‡ k 1 x ‡ k 2 x 2 ‡:
By inserting this into Eq. (2.31) we ®nd that the second solution is of the form
y 2 …x† ˆy 1 …x† ln x ‡ x r X1
mˆ1
A m x m :
…2:33†
Case 3 Roots di€ering by an integer
If the roots r 1 and r 2 of the indicial equation (2.29) di€er by an integer, say, r 1 ˆ r
and r 2 ˆ r p, where p is a positive integer, then we may always determine one
solution as before, namely, the solution corresponding to r 1 :
y 1 …x† ˆx r 1
…a 0 ‡ a 1 x ‡ a 2 x 2 ‡†:
To determine a second solution y 2 , we may proceed as in Case 2. The ®rst steps
are literally the same and yield Eq. (2.32). We determine 2r ‡ g 0 in Eq. (2.32).
Then from the indicial equation (2.29), we ®nd …r 1 ‡ r 2 †ˆg 0 1. In our case,
r 1 ˆ r and r 2 ˆ r p, therefore, g 0 1 ˆ p 2r. Hence in Eq. (2.32) we have
2r ‡ g 0 ˆ p ‡ 1, and we thus obtain
u 00
u 0 ˆ p ‡ 1
x
‡ :
Integrating, we ®nd
ln u 0 ˆ…p ‡ 1† ln x ‡ or u 0 ˆ x …p‡1† e …...† ;
where the dots stand for some series of positive powers of x. By expanding the
exponential function as before we obtain a series of the form
u 0 ˆ 1
x p‡1 ‡ k 1
x p ‡‡k p
x ‡ k p‡1 ‡ k p‡2 x ‡:
91