Airy functions
/ AIR-ee /
Imagine a wave that, as you move along, gradually runs out of energy: on one side it oscillates merrily, and on the other it has no room to oscillate and instead just decays away. The boundary between 'wave allowed' and 'wave forbidden' is governed by the Airy functions — the solutions of the deceptively simple equation y'' - x y = 0, which is about the simplest variable-coefficient equation imaginable, yet has no elementary solution.
The equation y'' = x y has the remarkable feature that the sign of the coefficient flips at x = 0. For x < 0 the equation behaves like y'' = -(positive) y, which is oscillatory (like y'' + y = 0), so solutions wiggle. For x > 0 it behaves like y'' = +(positive) y, which is exponential (like y'' - y = 0), so solutions grow or decay. The two standard solutions are the Airy function of the first kind, Ai(x), which decays gracefully to zero as x increases through the positive side and is the physically relevant 'bounded' one, and the second kind, Bi(x), which blows up there. Notice x = 0 is actually an ordinary point, so a power series works fine; the interest is not in any singularity but in this turning-point behaviour where oscillation hands over to decay.
Airy functions are the universal local description of a 'turning point' — the place in a quantum-mechanical problem where a particle reaches the edge of the region it is allowed to be, or in optics where a light ray grazes a caustic and produces the bright fringe of a rainbow. Wherever a smoothly varying coefficient passes through zero and the character of the solution switches from wavy to decaying, you will find an Airy function describing the crossover, and that universality is why so simple an equation earns a named pair of functions.
Solve y'' = x y by a power series about the ordinary point x = 0: matching coefficients gives a_(n+3) = a_n / ((n+3)(n+2)), a three-step recurrence. Starting from a_0 builds one solution, from a_1 the other; the specific combination that stays bounded as x grows is Ai(x), with the famous value Ai(0) = 1/(3^(2/3) Gamma(2/3)).
x = 0 is an ordinary point, so a plain power series works; the three-step recurrence comes from the lone x y term.
Do not expect Airy's simplicity to mean elementary solutions — y'' - x y = 0 has none in terms of powers, exponentials, or trig functions; the simplicity is in the equation, not the answer.