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# 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.

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## Mathematics Subject Classification

35-00*no label found*34-00

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