I need help with question 2 on this sheet.

1. Consider the function sinc f(I) = 1 1=0 Use the Maclaurin Series of sina to find a power series representation of f(r) centered at ro = 0. Conclude that f(r) is an analytic function. The function sing is often redefined to be the function f(x) so that is can be treated as an analytic function. This is an example where dividing an analytic function by a polynomial that is not non-zero still yields an analytic function. 2. Consider the differential equation (x] + 1)y" + 2ry' = 0. (a) Find all singular points. Recall that singular points can be complex numbers. (b) Since ro = 0 is an ordinary point, there are two linearly independent power series solutions (See Theorem 6.1.2). Use Theorem 6.1.2 to find a lower bound for the radius of convergence of the power series solutions. (c) One solution to the differential equation is given by y1 (z) = arctan = = ( - 1)" an+1 2n + 1 n=0 Find the radius of convergence of this power series. The radius of convergence should agree with the minimum for the radius of convergence you found in part (b). This is an instance where, even though the singular points are not real, the radius of convergence of the series solution is no more than the distance from the ordinary point we centered the power series at, To = 0, to the nearest singular point in the complex plane. (d) Convince yourself that another linearly independent solution is given by yz(r) = 1. What is the radius of convergence of this power series solution? This is an instance where the radius of convergence is larger than the minimum found via Theorem 6.1.2. (e) For old time's sake, what is the general solution for this differential equation? (f) (I will not grade this part of the problem, but I will include the solution in the answer key) Start from scratch: find two power series solutions centered at Zo = 0 for the differential equation (x2 + 1)y" + 2xy' = 0. You know what you will get, yz(x) = 1 and y1(z) = ()" 2n+1 1=0 - 2n + 1