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.
1 answer:
You might be interested in
You do what is asked for in the parentheses then multiply by the number outside of them.
For this case we must factor the following expression:
We rewrite the middle term as a sum of two terms whose product is
And whose sum is
These numbers are:
-5 and -3
So:
We factor the maximum common denominator of each group:
We take common factor
Answer: