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Airy Differential Equation

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Some authors define a general Airy differential equation as

y^('')+/-k^2xy=0.
(1)

This equation can be solved by series solution using the expansions

y=sum_(n=0)^(infty)a_nx^n
(2)
y^'=sum_(n=0)^(infty)na_nx^(n-1)
(3)
=sum_(n=1)^(infty)na_nx^(n-1)
(4)
=sum_(n=0)^(infty)(n+1)a_(n+1)x^n
(5)
y^('')=sum_(n=0)^(infty)(n+1)na_(n+1)x^(n-1)
(6)
=sum_(n=1)^(infty)(n+1)na_(n+1)x^(n-1)
(7)
=sum_(n=0)^(infty)(n+2)(n+1)a_(n+2)x^n.
(8)

Specializing to the "conventional" Airy differential equation occurs by taking the minus sign and setting k^2=1. Then plug (8) into

y^('')-xy=0
(9)

to obtain

sum_(n=0)^infty(n+2)(n+1)a_(n+2)x^n-xsum_(n=0)^inftya_nx^n=0
(10)
sum_(n=0)^infty(n+2)(n+1)a_(n+2)x^n-sum_(n=0)^inftya_nx^(n+1)=0
(11)
2a_2+sum_(n=1)^infty(n+2)(n+1)a_(n+2)x^n-sum_(n=1)^inftya_(n-1)x^n=0
(12)
2a_2+sum_(n=1)^infty[(n+2)(n+1)a_(n+2)-a_(n-1)]x^n=0.
(13)

In order for this equality to hold for all x, each term must separately be 0. Therefore,

a_2=0
(14)
(n+2)(n+1)a_(n+2)=a_(n-1).
(15)

Starting with the n=3 term and using the above recurrence relation, we obtain

5·4a_5=20a_5=a_2=0.
(16)

Continuing, it follows by induction that

a_2=a_5=a_8=a_(11)=...a_(3n-1)=0
(17)

for n=1, 2, .... Now examine terms of the form a_(3n).

a_3=(a_0)/(3·2)
(18)
a_6=(a_3)/(6·5)=(a_0)/((6·5)(3·2))
(19)
a_9=(a_6)/(9·8)=(a_0)/((9·8)(6·5)(3·2)).
(20)

Again by induction,

a_(3n)=(a_0)/([(3n)(3n-1)][(3n-3)(3n-4)]...[6·5][3·2])
(21)

for n=1, 2, .... Finally, look at terms of the form a_(3n+1),

a_4=(a_1)/(4·3)
(22)
a_7=(a_4)/(7·6)=(a_1)/((7·6)(4·3))
(23)
a_(10)=(a_7)/(10·9)=(a_1)/((10·9)(7·6)(4·3)).
(24)

By induction,

a_(3n+1)=(a_1)/([(3n+1)(3n)][(3n-2)(3n-3)]...[7·6][4·3])
(25)

for n=1, 2, .... The general solution is therefore

y=a_0[1+sum_(n=1)^infty(x^(3n))/((3n)(3n-1)(3n-3)(3n-4)...3·2)] +a_1[x+sum_(n=1)^infty(x^(3n+1))/((3n+1)(3n)(3n-2)(3n-3)...4·3)].
(26)

For a general k^2 with a minus sign, equation (◇) is

y^('')-k^2xy=0,
(27)

and the solution is

y(x)=1/3sqrt(x)[AI_(-1/3)(2/3kx^(3/2))-BI_(1/3)(2/3kx^(3/2))],
(28)

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

y(x)=A^'Ai(k^(2/3)x)+B^'Bi(k^(2/3)x).
(29)

If the plus sign is present instead, then

y^('')+k^2xy=0
(30)

and the solutions are

y(x)=1/3sqrt(x)[AJ_(-1/3)(2/3kx^(3/2))+BJ_(1/3)(2/3kx^(3/2))],
(31)

where J(z) is a Bessel function of the first kind.

A generalization of the Airy differential equation is given by

y^(''')-4xy^'-2y=0,
(32)

which has solutions

y=C_1[Ai(x)]^2+C_2Ai(x)Bi(x)+C_3[Bi(x)]^2
(33)

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

See also

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.

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Airy Differential Equation

Cite this as:

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

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