ASSIGNMENT 8 There are two parts to this assignment. The first part consists of questions on WeBWorK — the link is available on the course webpage. The second part consists of the questions on this page. You are expected to provide full solutions with complete justifications. You will be graded on the mathematical, logical and grammatical coherence and elegance of your solutions. Your solutions must be typed, with your name and student number at the top of the first page. If your solutions are on multiple pages, the pages must be stapled together. Your written assignment must be handed in before the start of your recitation on Friday, March 27. The online assignment will close at 9:00 a.m. on Friday, March 27. 1. Find a power series expression, and its interval of convergence, for the function f (x) = 1 . 1 + 4x2 2. In a recent workshop, you examined the Bessel function J(x) = x ν X 2 n≥0 (−1)n x2n , + ν + 1) 22n n!Γ(n where ν is a positive constant. Prove that this function satisfies the differential equation x2 J 00 (x) + xJ 0 (x) + (x2 − ν 2 )J(x) = 0. You may use without justification the identity Γ(m + 1) = mΓ(m) for m > 0 (recall that you were asked to prove this in the workshop and on the redemption test). 3. Consider the differential equation f 0 (x) = 2xf (x) + 4x. (a) Find a power series f (x) = X an xn satisfying this differential equation. n≥0 (b) Write down a closed form expression for the power series you found in part (a).