(a) Use the result of Problem 34 to express the general solution of Airy’s differential equation for x > 0 in terms of Bessel functions.

(b) Verify the results in part (a) using (18).

Problem 34

Show that y = x^1/2 w(2/3 αx^3/2) is a solution of Airy’s differential equation y'' + α²xy = 0, x > 0, whenever w is a solution of Bessel’s equation of order 1/3, that is, t²w'' + tw' + (t² – 1/9)w = 0, t > 0. After differentiating, substituting, and simplifying, then let t = 2/3 αx^3/2.