Find the first four nonzero terms in each of two power series solutions about th

Question

Find the first four nonzero terms in each of two power series solutions about the origin.

(e^-x)y'' + ln(1+x)y' -xy =0

Any help would be appreciated. Thanks!

Explanation / Answer

ANSWER this will help you For y = e^(1/x), as x -> 0 from the right we see that y -> e^(infinity) = infinity. As x -> 0 from the left y -> e^(-infinity) = 0. As x -> infinity, y goes to 0, and as x -> - infinity y -> e^0 = 1. So in summary, there is a vertical asymptote at x = 0, (for x -> 0 from the right), a horizontal asymptote at y = 0, (for x -> infinity) and a horizontal asymptote at y = 1, (for x -> - infinity). I'll just have a look at the other one now.... For y = ln(1 + e^x), the first thing to note is that this is defined for all real numbers, and hence there are no vertical asymptotes. As x -> - infinity we see that (1 + e^x) -> (1 + e^(-infinity)) = 1, and since ln(1) = 0 there is a horizontal asymptote at y = 0 as x -> - infinity. As x -> infinity we see that (1 + e^x) -> e^x, and since ln(e^x) = x there is a slant asymptote of y = x as x goes to infinity.

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