TITLE: Initial Value Problem for $y''+xy'+x^2y=0$
QUESTION [3 upvotes]: Does anyone have a solution for the initial value problem: $$y''+xy'+x^2y=0, y(0)=1, y'(0)=1 ?$$
I try power series solution but I have trouble finding a pattern for the general term of the series.

REPLY [1 votes]: The solution of $$h'+xh=-h^2-x^2, h(0)=1$$ seems to be more complex than the solution of $$y''+xy'+x^2y=0, y(0)=1, y'(0)=1$$ as shown by doraemonpaul. I think that the solution you obtained earlier with power series and the nice recurrence relation is probably the best way to go.
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