connection between Riccati equation and Airy functions

We report an interesting connection relating Riccati equation with Airy functions. Let us consider the nonlinear complex operator 𝔏:zζ with kernel given by

dζdz+ζ2+a(z)ζ+b(z)=0, (1)

a nonlinear ODE of the first order so-called Riccati equation. In order to accomplish our purpose we particularize (1) by setting a(z)0 and b(z)=-z. Thus (1) becomes

dζdz+ζ2=z. (2)

(2) can be reduced to a linear equation of the second order by the suitable change: ζ=w(z)/w(z), whence


which leads (2) to

w′′-zw=0. (3)

Pairs of linearly independentMathworldPlanetmath solutions of (3) are the Airy functions.

Title connection between Riccati equation and Airy functions
Canonical name ConnectionBetweenRiccatiEquationAndAiryFunctions
Date of creation 2013-03-22 18:09:07
Last modified on 2013-03-22 18:09:07
Owner perucho (2192)
Last modified by perucho (2192)
Numerical id 5
Author perucho (2192)
Entry type Derivation
Classification msc 35-00
Classification msc 34-00