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[parent] connection between Riccati equation and Airy functions (Derivation)

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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Cross-references: solutions, linearly independent, second order, linear equation, reduced, order, first order, ODE, kernel, operator, complex, functions, Riccati equation, connection

This is version 2 of connection between Riccati equation and Airy functions, born on 2008-06-17, modified 2008-06-17.
Object id is 10708, canonical name is ConnectionBetweenRiccatiEquationAndAiryFunctions.
Accessed 503 times total.

Classification:
AMS MSC34-00 (Ordinary differential equations :: General reference works )
 35-00 (Partial differential equations :: General reference works )

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