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You can use the ore_algebra package as follows.
First, you need to install the latest version (not the packaged one which is not up-to-date), from the shell, just type:
sage -pip install --upgrade --user git+https://github.com/mkauers/ore_algebra.git
Then, in Sage:
sage: from ore_algebra import *
sage: R.<x> = PolynomialRing(ZZ); A.<Dx> = OreAlgebra(R)
sage: O = Dx^2 + x^2*Dx + 1
sage: O
Dx^2 + x^2*Dx + 1
sage: O.power_series_solutions(n=10)
[x - 1/6*x^3 - 1/12*x^4 + 1/120*x^5 + 7/360*x^6 + 13/1680*x^7 - 11/10080*x^8 - 209/120960*x^9 + O(x^10),
1 - 1/2*x^2 + 1/24*x^4 + 1/20*x^5 - 1/720*x^6 - 13/2520*x^7 - 179/40320*x^8 + 17/90720*x^9 + O(x^10)]
