How to test if a certain differential equation has an analytic solution (solution has Taylor expansion)?

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How to test if the differential equation

$$(1+x)y'''+y y''-y'^2+(2+x)y''+y'=0,$$ $$y(0)=1, y'(0)=1, y'(\infty)=0$$

Has an analytic solution? Solution that equals its Taylor series?