(a) Find two power series solutions for y'' + xy' + y = 0 and express the solutions y₁(x) and y₂(x) in terms of summation notation.

(b) Use a CAS to graph the partial sums S_N (x) for y₁(x). Use N = 2, 3, 5, 6, 8, 10. Repeat using the partial sums S_N (x) for y₂(x).

(c) Compare the graphs obtained in part (b) with the curve obtained by using a numerical solver. Use the initial-conditions y₁(0) = 1, y'₁(0) = 0, and y₂(0) = 0, y'₂(0) = 1.

(d) Reexamine the solution y₁(x) in part (a). Express this series as an elementary function. Then use (5) of Section 4.2 to find a second solution of the equation. Verify that this second solution is the same as the power series solution y₂(x).