Determine a recurrence relation for the coefficients in the power series about 20 = 0 for the general solution of (1 - x2 )y" ty'ty= xe". Use this to write the first five nonzero terms (i.e., all terms up to order x inclusive) of the general solution. 2. Series solutions can be used to solve differential equations of any order, though this is not always the best method. (a) Consider the first order differential equation y' + xy =0 with initial condition y(1) = 1. Write the solution as a series y = _ an(x - 1)" expanded about x = 1. Find n=0 the recurrence relation for the coefficients an and write the first 4 nonzero terms. (b) . Solve this problem by earlier (Chapter 2) methods, using that this DE is either separable or linear. 3. (a) Find all singular points in the complex plane for the equation 2(2 + x2)xy" + xy' + 3x-y = 0. (b) If one attempts to solve the initial value problem 2(2 + x2 )y" + y' + 3xy = 0, y(1) = 0, y'(1) = 1, by an expansion of the form y = _ an(x - 1)", use the result of part n=0 (a) to find a lower bound for the radius of convergence of the solution