Attached are two problems, one with an answer that I don't understand how it was derived and one problem without the answer that I would like to see how it is solved.
Power Series Methods - Introduction and Review of Power Series
14. Find two linearly independent power series solutions of the given differential equation. Determine the radius of the convergence of each series, and identify the general solution in terms of familiar elementary functions.
y" + y = x
y(x) = x + c0 (1 - x2/2! + x4/4! + x6/6! + ...) + (c1-1) (x - x3/3! + x5/5! + x7/7! + ...) = x + c0cosx + (c1-1) sinx ;  = 
24. Establish the binomial series in (12) by means of the following steps. (a) Show that y = (1 +x) satisfies the initial value problem (1 + x)y' = y, y(0) = 1. (b) Show that the power series method gives the binomial series in (12) as the solution of the initial value problem in part (a), and that this series converges if x < 1. (x) Explain why the validity of the binomial series given in (12) follows from parts (a) and (b).
(1 +x) = 1 + x + [( - 1)x2/2!] + [( - 1)( - 2)x3/3!] + ... (12)
Elementary differential equations and power series methods are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.