Finding linearly independent power series

Question

Assignment:

Power Series Methods - Introduction and Review of Power Series

Q1. 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

Q2. Establish the binomial series in (12) by means of the following steps.  (a) Show that y = (1 +x)^a satisfies the initial value problem (1 + x)y' = ay, 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 ½x½ < 1.  (x) Explain why the validity of the binomial series given in (12) follows from parts (a) and (b).

 (1 +x)^a = 1 + ax + [a(a - 1)x²/2!] + [a(a - 1)a(a - 2)x³/3!]

Provide complete and step by step solution for the question and show calculations and use formulas.