Can Variable Coefficients and Nonhomogeneous Terms Be Solved Using Power Series?

[rumy]

hello everybody,
i have a different DE question, actually i have searced it in the net but didn't find the answer.
what if there is a DE with variable coefficients (needed power series sol'n)
but a NONHOMOGENEOUS one? (actually it's kinda urgent, have only one day)
for example: x^2 y'' + xy' + (x+1)y = 1/x^2
my sol'n:
actually i can't think of anything, if the righthand side was 0 it could be solved by
y=SUM( an * x^(n+r) ) then apply the well known solution (the long one, can't be written here, it is very long)
but i don't know even which method to apply to solve a nonhomo
by the way if the general solution i need is y = yc + yp
yc: y complementary
yp: y particular
i need yp (don't know which method to use)

 

[HallsofIvy]

Write the righthand side as x^-2 and fit it into your series!

Or, if you don't like that negative power, multiply the entire equation by x^2 and write it as x^4 y"+ x^3y'+ x^2(x+1)y= 1. Include the 1 in your series for y.