I am now trying to understand what the so-called Stokes phenomenon means. In this page Stokes phenomenon, it reads that
For large $x$ of given argument the solution (of the Airy equation) can be approximated by a linear combination of the functions $$ \frac{e^{\pm \frac{2}{3} x^{3/2}}}{x^{1/4}}. $$
The question is, these two terms apparently differ by orders --- one is exponentially big while the other exponentially small. Thus, possibly there are some terms in-between? Apparently, it is meaningful to approximate the original function by a linear combination of the two above only if the two above are the 1st and 2nd order terms in the asymptotic expansion. This would mean, the leading term can approximate the original function with an exponentially small error. Is it really so?
No, it's not true. As $x \to +\infty$ the solution $Ai(x)$ satisfies $$ Ai(x) \sim \dfrac{e^{-2 x^{3/2}/3}}{\sqrt{\pi}} \left( \dfrac{1}{2 x^{1/4}} - \dfrac{5}{96\; x^{7/4}} + \dfrac{385}{9216\; x^{13/4}} - \ldots \right) $$ while the other solution $Bi(x)$ satisfies $$ Bi(x) \sim \dfrac{e^{2 x^{3/2}/3}}{\sqrt{\pi}} \left( \dfrac{1}{x^{1/4}} + \dfrac{5}{48\; x^{7/4}} + \dfrac{385}{4608\; x^{13/4}} + \ldots \right)$$