Mathematical Methods for Physicists: A concise introduction - Site Map

SOLUTIONS IN POWER SERIES
indicial equation di€er by an integer. We now take a general look at these cases.
For this purpose, let us consider the following di€erential equation which is highly
important in mathematical physics:
x 2 y 00 ‡ xg…x†y 0 ‡ h…x†y ˆ 0;
…2:27†
where the functions g…x† and h…x† are analytic at x ˆ 0. Since the coecients are
not analyic at x ˆ 0, the solution is of the form
y…x† ˆx r X1
mˆ0
We ®rst expand g…x† and h…x† in power series,
a m x m …a 0 6ˆ 0†: …2:28†
g…x† ˆg 0 ‡ g 1 x ‡ g 2 x 2 ‡
h…x† ˆh 0 ‡ h 1 x ‡ h 2 x 2 ‡:
Then di€erentiating Eq. (2.28) term by term, we ®nd
y 0 …x† ˆX1
mˆ0
…m ‡ r†a m x m‡r1 ;
y 00 …x† ˆX1
mˆ0
…m ‡ r†…m ‡ r 1†a m x m‡r2 :
By inserting all these into Eq. (2.27) we obtain
x r ‰r…r 1†a 0 ‡Š‡…g 0 ‡ g 1 x ‡†x r …ra 0 ‡†
‡…h 0 ‡ h 1 x ‡†x r …a 0 ‡ a 1 x ‡†ˆ0:
Equating the sum of the coecients of each power of x to zero, as before, yields a
system of equations involving the unknown coecients a m . The smallest power is
x r , and the corresponding equation is
Since by assumption a 0 6ˆ 0, we obtain
‰r…r 1†‡g 0 r ‡ h 0 Ša 0 ˆ 0:
r…r 1†‡g 0 r ‡ h 0 ˆ 0 or r 2 ‡…g 0 1†r ‡ h 0 ˆ 0: …2:29†
This is the indicial equation of the di€erential equation (2.27). We shall see that
our series method will yield a fundamental system of solutions; one of the solutions
will always be of the form (2.28), but for the form of other solution there will
be three di€erent possibilities corresponding to the following cases.
Case 1 The roots of the indicial equation are distinct and do not di€er by an
integer.
Case 2 The indicial equation has a double root.
Case 3 The roots of the indicial equation di€er by an integer.
We now discuss these cases separately.
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