Some authors define a general Airy differential equation as
|
(1) |
This equation can be solved by series solution using the expansions
Specializing to the ``conventional'' Airy differential equation occurs by taking the Minus Sign and setting
. Then plug (4) into
|
(5) |
to obtain
|
(6) |
|
(7) |
|
(8) |
|
(9) |
In order for this equality to hold for all , each term must separately be 0. Therefore,
Starting with the term and using the above Recurrence Relation, we obtain
|
(12) |
Continuing, it follows by Induction that
|
(13) |
for , 2, .... Now examine terms of the form .
Again by Induction,
|
(17) |
for , 2, .... Finally, look at terms of the form ,
By Induction,
|
(21) |
for , 2, .... The general solution is therefore
|
|
|
|
|
(22) |
For a general with a Minus Sign, equation (1) is
|
(23) |
and the solution is
|
(24) |
where is a Modified Bessel Function of the First Kind. This is usually expressed in terms of the Airy
Functions
and
|
(25) |
If the Plus Sign is present instead, then
|
(26) |
and the solutions are
|
(27) |
where is a Bessel Function of the First Kind.
See also Airy-Fock Functions, Airy Functions, Bessel Function of the First Kind, Modified Bessel
Function of the First Kind
© 1996-9 Eric W. Weisstein
1999-05-25