 TOPICS  # Airy Differential Equation

Some authors define a general Airy differential equation as (1)

This equation can be solved by series solution using the expansions   (2)   (3)   (4)   (5)   (6)   (7)   (8)

Specializing to the "conventional" Airy differential equation occurs by taking the minus sign and setting . Then plug (8) into (9)

to obtain (10) (11) (12) (13)

In order for this equality to hold for all , each term must separately be 0. Therefore,   (14)   (15)

Starting with the term and using the above recurrence relation, we obtain (16)

Continuing, it follows by induction that (17)

for , 2, .... Now examine terms of the form .   (18)   (19)   (20)

Again by induction, (21)

for , 2, .... Finally, look at terms of the form ,   (22)   (23)   (24)

By induction, (25)

for , 2, .... The general solution is therefore (26)

For a general with a minus sign, equation (◇) is (27)

and the solution is (28)

where is a modified Bessel function of the first kind. This is usually expressed in terms of the Airy functions and  (29)

If the plus sign is present instead, then (30)

and the solutions are (31)

where is a Bessel function of the first kind.

A generalization of the Airy differential equation is given by (32)

which has solutions (33)

(Abramowitz and Stegun 1972, p. 448; Zwillinger 1997, p. 128).

Airy-Fock Functions, Airy Functions, Bessel Function of the First Kind, Modified Bessel Function of the First Kind

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## References

Abramowitz, M. and Stegun, I. A. (Eds.). "Airy Functions." §10.4.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 446-452, 1972.Zwillinger, D. (Ed.). CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 413, 1995.Zwillinger, D. Handbook of Differential Equations, 3rd ed. Boston, MA: Academic Press, p. 121, 1997.

## Referenced on Wolfram|Alpha

Airy Differential Equation

## Cite this as:

Weisstein, Eric W. "Airy Differential Equation." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AiryDifferentialEquation.html