# 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 $\mathfrak{L}:z\in\mathbb{C}\mapsto\zeta$ with kernel given by

 $\frac{d\zeta}{dz}+\zeta^{2}+a(z)\zeta+b(z)=0,$ (1)

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

 $\frac{d\zeta}{dz}+\zeta^{2}=z.$ (2)

(2) can be reduced to a linear equation of the second order by the suitable change: $\zeta=w^{\prime}(z)/w(z)$, whence

 $\zeta^{\prime}=\frac{w^{\prime\prime}}{w}-\frac{w^{\prime 2}}{w^{2}},\qquad% \zeta^{2}=\left(\frac{w^{\prime}}{w}\right)^{2},$

which leads (2) to

 $w^{\prime\prime}-zw=0.$ (3)

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

Title connection between Riccati equation and Airy functions ConnectionBetweenRiccatiEquationAndAiryFunctions 2013-03-22 18:09:07 2013-03-22 18:09:07 perucho (2192) perucho (2192) 5 perucho (2192) Derivation msc 35-00 msc 34-00