MHB Analyzing Local Behavior at x→0+ of y'+xy=1/x^3

[meenums34]

Asked to find first three terms in the local behavior as x→0+of the solutions of
y′+xy=1/x^3

This was taken by bender and orszag book

Working :
I tried to use method of dominance but later realized that we can find using the series expansion. But I am not sure how to proceed. Please advise how to proceed with it.

 

[HallsofIvy]

Write y as \sum a_nx^n where n runs over all integers, both negative and positive. Then y'= \sum na_nx^{n-1} and xy= \sum a_nx^{n+1}.

The equation becomes \sum na_nx^{n-1}+ \sum a_nx^{n+1}= x^{-3}.
In the first sum let j= n-1 so that we have \sum (j+1)a_{j+1}x^j.
In the second sum let j= n+ 1 so that we have \sum a_{j-1}x^j.

Now the equation is \sum (j+1)a_{j+1}x^j+ \sum a_{j-1}x^j= \sum ((j+1)a_{j+1}+ a_{j- 1})x^j= x^{-3}.

Since power series expansion is unique, we have the (infinite) set of equations
(j+1)a_{j+1}+ a_{j-1}= 0 for all j except j= -3 and
(-3+ 1)a_{-3+ 1}+ a_{-3-1}= -2a_{-2}+ a_{-4}= 1.