The discussion following Example suggests that the differential equation y″ + y = 0 coul...

The discussion following Example suggests that the differential equation y″ + y = 0 could be used to introduce and define the familiar sine and cosine functions. In a similar fashion, the Airy equation

y″ = xy serves to introduce two new special functions that appear in applications ranging from radio waves to molecular vibrations. Derive the first three or four terms of two different power series solutions of the Airy equation. Then verify that your-results agree with the formulas

and

for the solutions that satisfy the initial conditions y1 (0) = 1, y1′(0) = 0 and y2(0) = 0, y2′(O) = 1, respectively. The special combinations

and

define the standard Aiiy functions that appear in mathematical tables and computer algebra systems. Their graphs shown in Fig. 3.2.3 exhibit trigonometric-lilce oscillatory behavior for x < 0, whereas Ai(x) decreases exponentially and Bi(x) increases exponentially as x → ∞. It is interesting to use a computer algebra system to investigate how many terms must be retained in the y1 − and y2 − series above to produce a figure that is visually indistinguishable from Fig. 3.2.3 (which is based on high-precision approximations to the Airy functions).