Question

5. The differential equation y 00 − xy = 0 is called Airy’s equation, and is used in physics to model the refraction of light. (a) Assume a power series solution, and find the recurrence relation of the coefficients. [Hint: When shifting the indices, one way is to let m = n − 3, then factor out x n+1 and find an+3 in terms of an. Alternatively, you can find an+2 in terms of an−1.] (b) Show that a2 = 0. [Hint: the two series for y 00 and xy don’t “start” at the same power of x, but for any solution, each term must be zero. (Why?)] (c) Find the particular solution when y(0) = 1, y 0 (0) = 0, as well as the particular solution when y(0) = 0, y 0 (0) = 1.

Answer 1

Not sure why, but I wasn't able to post my solution as text, so I've written it elsewhere and am posting screenshots of it here.

In the fifth attachment, the first solution is shown above the second one.