(a) Coefficient formulas, show how a and b in (1) can be expressed in terms of 1...

Question:

(a) Coefficient formulas, show how a and b in (1) can be expressed in terms of λ1 and λ2. Explain how these formulas can be used in constructing equations for given bases.
(b) Root zero, solve y" + 4y" = 0 (i) by the present method, and (ii) by reduction to first order. Can you explain why the result must be the same in both cases? Can you do the same for a general ODE y" + ay' = 0?
(c) Double root, verify directly that xeλx with λ = – a/2 is a solution of (1) in the case of a double root. Verify and explain why y = e–2x is solution of y" – y' – 6y = 0 but xe–2x is root.
(d) Limits double roots should be limiting cases of distinct roots λ1, λ1 as, say, λ2 → λ1. Experiment with this idea. (Remember 1’Hopital’s rule from calculus.) Can you arrive at xeλ1x? Give it a try.