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 $𝔏:z\in ℂ\mapsto \zeta$ 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

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

which leads (2) to

$$w^{\prime \prime }-zw=0.$$(3)

Pairs of linearly independent solutions of (3) are the Airy functions.