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.