Let us analyze Airy's equation  y''= xy. This equation arises in many different contexts, most commonly as a limiting case of Schrodinger's equation.

For x near zero, the approximate equation is y''=0, and this has the general solution y=

The solution agrees with the simple approximation on a decently large interval about zero. Eyeballing it, we can't tell the difference between them on [-.2, .4].

Thus is "good enough" means 5%, then this approximation is good enough on the interval (-.8,.8)

Now let's turn to x very large negative. If x ~ -, then we can try to approximate with y —�. We'll try ω=5. For starters, set c1=1 (no scaling), c2=0 (no phaseshift)

We see that the Airy function does have approximately the same period, and that we can scale (c1=.55) and phase shift (c2=-2.7)

Pretty neat huh?  What is the error?

Let's see if this solution is good on bigger intervals:

Now it is starting to get away from us, mainly because the period is changing. What else changes towards the end points of the intervals?

Do you think that for higher frequencies you will get a better or worse approximation (meaning: valid on a longer or shorter interval)? Guess, and experiment.

Now let's go to the other extremem, x ~ , then we can try to approximate with y —�. Let's use again k=5.

Well, our approximation is off by many orders of magnitude, so let's start fiddling with the knobs......

There the shapes look to be about the same. Now let's twiddle the other dial....

Not bad.

Let's solve the equation for a variety of initial conditions.