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connection between Riccati equation and Airy functions

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\begin{document}
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
\begin{equation}
\frac{d\zeta}{dz}+\zeta^2+a(z)\zeta+b(z)=0,
\end{equation}
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
\begin{equation}
\frac{d\zeta}{dz}+\zeta^2=z.
\end{equation}
(2) can be reduced to a linear equation of the second order by the suitable change: $\zeta=w'(z)/w(z)$, whence
\begin{equation*} 
\zeta'=\frac{w''}{w}-\frac{w'^2}{w^2}, \qquad \zeta^2=\left(\frac{w'}{w}\right)^2,
\end{equation*}
which leads (2) to
\begin{equation}
w''-zw=0.
\end{equation}
Pairs of linearly independent solutions of (3) are the Airy functions.
  

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