# 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 $\U0001d50f:z\in \u2102\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

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

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 |
---|---|

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 |