A Higher Order Non–Linear Differential Equation and a Generalization of the Airy Function

In this paper a higher order non–linear differential equation is given and it becomes a higher order Airy equation (in our terminology) under the Cole–Hopf transformation. For the even case a solution is explicitly constructed, which is a generalization of the Airy function. We start with an example to state our motivation. For the simple non–linear equation of Riccati type y + y = x (1) with a smooth function y = y(x), we apply the Cole–Hopf transformation y = d dx log u = u u (u = u(x)) (2) to (1). Then we obtain the famous Airy equation [1] u = xu. (3) ∗E-mail address : [email protected] 1 This equation plays an important role in both Quantum Optics and Mathematical Physics, so many studies have been made. See for example [2]. This has two well–known solutions called the Airy function Ai(x) and the Airy function of the second kind Bi(x). For the details see [3]. In particular, Ai(x) is written in terms of an improper integral u(x) = 1 π ∫ ∞