The Bright Edge of a Rainbow

Light near the edge of a rainbow obeys an equation with no formula answer. So build the answer as an infinite polynomial, one term at a time, letting the equation choose every coefficient. A few terms already match it well nearby.

How it works

Series solutions

In one sentence

Near an ordinary point, $y'' + p(x)y' + q(x)y = 0$ has power series solutions $y = \sum a_n(x - x_0)^n$ , found by substituting and matching coefficients; for $y'' = xy$ they converge for all $x$ .

General formula

{\htmlData{sym=0}{y''}} = {\htmlData{sym=1}{x}}\,{\htmlData{sym=2}{y}} \;\Rightarrow\; {\htmlData{sym=2}{y}} = 1 + \frac{{\htmlData{sym=1}{x}}^3}{2\cdot3} + \frac{{\htmlData{sym=1}{x}}^6}{2\cdot3\cdot5\cdot6} + \cdots = 1 + \sum_{k=1}^{\infty}\frac{{\htmlData{sym=1}{x}}^{3k}}{\prod_{j=1}^{k}(3j-1)(3j)}

Once the first two coefficients are chosen, the equation forces every later one, so the series is the solution, built term by term.

What the symbols mean

$y''$
Curvature
How sharply the amplitude bends.
$x$
Position
Distance across the edge of the bright band.
$y$
Amplitude
The light wave's strength at x.

The Airy equation $y'' = xy$ has no elementary solution, so assume a power series $y = \sum a_nx^n$ and substitute.
Matching powers gives $a_{n+2} = \frac{a_{n-1}}{(n+2)(n+1)}$ : each coefficient comes from the one three places earlier.
Starting from $a_0 = 1$ and $a_1 = 0$ gives the series above; it converges for every $x$ .
For $x < 0$ the solution oscillates and for $x > 0$ it grows. That switch is the bright edge of a rainbow.

Learn the math

Series SolutionsDiff Eq · Lesson 37, Series Solutions