I've been told that the following equation
$$\left(x-\frac{\partial^2}{\partial x^2} \right) y =1$$
has a solution of the form
$$y(x)=\int_0^\infty e^{-t^3/3 - ixt}dt$$
(an exponential of something, but I can't find what's in the exponent)
What's the solution to this equation in terms of the integral of an exponential? $y(x)$ has to vanish as $x\rightarrow \infty$, so I guess the term corresponding to the Bi function is not present.
x is complex
The Airy integrals that defines the Airy functions , $\text{Ai}$ and $\text{Bi}$ are
$$ \text{Ai}(x)=\frac{1}{\pi}\int_0^{\infty}\cos\left(xt+\frac{t^3}{3} \right)~dt\\ \text{Bi}(x)=\frac{1}{\pi}\int_0^{\infty}\sin\left(xt+\frac{t^3}{3} \right)~dt+\frac{1}{\pi}\int_0^{\infty}\exp\left(xt+\frac{t^3}{3} \right)~dt $$
and Airy's differential equation and solution are
$$ \frac{d^2f}{dx^2}-xf=0\\ f=a\text{Ai}(x)+b\text{Bi}(x) $$
The Airy functions are periodic for negative $x$ and hyperbolic for positive $x$, with $\text{Ai}\to 0 ~\& ~\text{Bi}\to\infty \text{ as } x\to\infty.$