Airy equation

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The Airy equation is the second-order linear ordinary differential equation $y'' - xy = 0.$ It occurred first in G.B. Airy's research in optics [Ai]. Its general solution can be expressed in terms of Bessel functions of order $\pm 1/3$: $y(x) = c_1 \sqrt{x} J_{1/3}\left(\frac{2}{3}\mathrm{i}x^{3/2}\right) + c_2 \sqrt{x} J_{-1/3}\left(\frac{2}{3}\mathrm{i}x^{3/2}\right).$ Since the Airy equation plays an important role in various problems of physics, mechanics and asymptotic analysis, its solutions are regarded as forming a distinct class of special functions (see Airy functions).
The solutions of the Airy equation in the complex plane $z$, $w'' - zw = 0,$ have the following fundamental properties:
1) Every solution is an entire function of $z$ and can be expanded in a power series $w(z) = w(0) \left( 1 + \frac{z^3}{2.3} + \frac{z^6}{(2.3).(5.6)} + \cdots \right) + w'(0) \left( z + \frac{z^4}{3.4} + \frac{z^7}{(3.4).(6.7)} + \cdots \right),$ which converges for all $z$.
2) If $w(z) \not\equiv 0$ is a solution of the Airy equation, then so are $w(\omega z)$ and $w(\omega^2 z)$, where $w=\mathrm{e}^{2\pi\mathrm{i}/3}$, and any two of these solutions are linearly independent. The following identity holds: $w(z) + w(\omega z) + w(\omega^2 z) \equiv 0.$