“Airy Differential Equation”的概念、定义、习题解题方法及翻译-数学辞典

单词

Airy Differential Equation

释义

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)

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,

			(10)
			(11)

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

			(14)
			(15)
			(16)

Again by Induction,

(17)

for

, 2, .... Finally, look at terms of the form

			(18)
			(19)
			(20)

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 AiryFunctions

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 BesselFunction of the First Kind

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