Power Series

Find the first 4 nonzero terms in a power series expansion?

Find the first 4 nonzero terms in a power series expansion about x=0 for the general solution to
Y’’ – (x+1) y = 0

Notice that y[x] = AiryAi[1 + x]c1 + AiryBi[1 + x]c2, where c1 and c2 are arbitrary real numbers not both of which are zero. If c1 and c2 are not both zero, then the first four terms of the power series expansion of y[x] about x = 0 are given by

......(c1 AiryAi[1] + c2 AiryBi[1]) +

......(c1 AiryAiPrime[1] + c2 AiryBiPrime[1]) x +

......1/2 (c1 AiryAi[1] + c2 AiryBi[1]) x^2 +

......1/6 (c1 (AiryAi[1] + AiryAiPrime[1]) + c2 (AiryBi[1] + AiryBiPrime[1])) x^3

Some of the four terms may or may not be zero, depending on the values of c1 and c2.

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