Some authors define a general Airy differential equation as
![\begin{displaymath}
y''\pm k^2xy = 0.
\end{displaymath}](a_434.gif) |
(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
![\begin{displaymath}
y''-xy = 0
\end{displaymath}](a_443.gif) |
(5) |
to obtain
![\begin{displaymath}
\sum_{n=0}^\infty (n+2)(n+1) a_{n+2} x^n - x \sum_{n=0}^\infty a_n x^n = 0
\end{displaymath}](a_444.gif) |
(6) |
![\begin{displaymath}
\sum_{n=0}^\infty (n+2)(n+1) a_{n+2} x^n - \sum_{n=0}^\infty a_n x^{n+1} = 0
\end{displaymath}](a_445.gif) |
(7) |
![\begin{displaymath}
2a_2 + \sum_{n=1}^\infty (n+2)(n+1) a_{n+2} x^n - \sum_{n=1}^\infty a_{n-1} x^n = 0
\end{displaymath}](a_446.gif) |
(8) |
![\begin{displaymath}
2a_2 + \sum_{n=1}^\infty [(n+2)(n+1) a_{n+2}-a_{n-1}]x^n = 0.
\end{displaymath}](a_447.gif) |
(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
![\begin{displaymath}
5 \cdot 4 a_5 = 20 a_5 = a_2 = 0.
\end{displaymath}](a_452.gif) |
(12) |
Continuing, it follows by Induction that
![\begin{displaymath}
a_2 = a_5 = a_8 = a_{11} = \ldots a_{3n-1} = 0
\end{displaymath}](a_453.gif) |
(13) |
for
, 2, .... Now examine terms of the form
.
Again by Induction,
![\begin{displaymath}
a_{3n} = {{a_0} \over {[(3n)(3n-1)][(3n-3)(3n-4)]\cdots [6 \cdot 5][3 \cdot 2]}}
\end{displaymath}](a_461.gif) |
(17) |
for
, 2, .... Finally, look at terms of the form
,
By Induction,
![\begin{displaymath}
a_{3n+1} = {{a_1} \over {[(3n+1)(3n)][(3n-2)(3n-3)] \cdots [7 \cdot 6][4 \cdot 3]}}
\end{displaymath}](a_469.gif) |
(21) |
for
, 2, .... The general solution is therefore
|
|
|
|
|
(22) |
For a general
with a Minus Sign, equation (1) is
![\begin{displaymath}
y''-k^2xy = 0,
\end{displaymath}](a_473.gif) |
(23) |
and the solution is
![\begin{displaymath}
y(x)= {\textstyle{1\over 3}}\sqrt{x}\left[{AI_{-1/3}\left({{...
...I_{1/3}\left({{\textstyle{2\over 3}} kx^{3/2}}\right)}\right],
\end{displaymath}](a_474.gif) |
(24) |
where
is a Modified Bessel Function of the First Kind. This is usually expressed in terms of the Airy
Functions
and
![\begin{displaymath}
y(x) = A' \mathop{\rm Ai}\nolimits (k^{2/3}x)+B' \mathop{\rm Bi}\nolimits (k^{2/3}x).
\end{displaymath}](a_478.gif) |
(25) |
If the Plus Sign is present instead, then
![\begin{displaymath}
y''+k^2xy = 0
\end{displaymath}](a_479.gif) |
(26) |
and the solutions are
![\begin{displaymath}
y(x)= {\textstyle{1\over 3}}\sqrt{x}\left[{AJ_{-1/3}\left({{...
...J_{1/3}\left({{\textstyle{2\over 3}} kx^{3/2}}\right)}\right],
\end{displaymath}](a_480.gif) |
(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