determine a series if solution of the equation y"(x)+x.y'(x)+y(x)=0

Question

show me step plz

Answer 1

review the conditions for the series to exist. In this case, it will exist for x=0, so we want

y = ∑a_nx^n
y' = ∑na_nx^(n-1)
y" = ∑n(n-1)a_nx^(n-2)

The DE then says
∑n(n-1)a_nx^(n-2) + x*∑na_nx^(n-1) + ∑a_nx^n = 0

∑n(n-1)a_nx^(n-2) + ∑na_nx^n + ∑a_nx^n = 0

∑n(n-1)a_nx^(n-2) + ∑(n+1)a_nx^n = 0

Now shift y" so the powers of x add up

∑(n+2)(n+1)a_n+2x^n + ∑(n+1)a_nx^n = 0

∑[(n+2)(n+1)a_n+2 + (n+1)a_n]x^n = 0

For the series to be zero for all x, we have the recurrence relation

(n+2)(n+1)a_n+2 + (n+1)a_n = 0

a_n+2 = -a_n/(n+2)

At this point, just start listing the coefficients and see where you go. There is a similar DE used as an example at

http://tutorial.math.lamar.edu/Classes/DE/SeriesSolutions.aspx