Chapter 9
Series solutions of O.D.E.
(Frobenius’ method)
Let us look at the a very simple (ordinary)
differential equation,
|
(9.1) |
with initial conditions
,
. Let us assume that there is a solution that is
analytical near
. This means that near
the function has a Taylor’s series
|
(9.2) |
(We shall ignore questions of convergence.)
Let us proceed
Combining this together we have
Here we have collected terms of equal power
of
. The reason is simple. We are requiring a power series
to equal
. The only way that can work is if each power of
in the power series has zero coefficient. (Compare
a finite polynomial....) We thus find
|
(9.5) |
The last relation is called a recurrence of
recursion relation, which we can use to bootstrap from a given
value, in this case
and
. Once we know these two numbers, we can determine
,
and
:
|
(9.6) |
These in turn can be used to determine
, etc. It is not too hard to find an explicit
expression for the
’s
The general solution is thus
|
(9.8) |
The technique sketched here can be proven to
work for any differential equation
|
(9.9) |
provided that
,
and
are analytic at
. Thus if
,
and
have a power series expansion, so has
.