How to know if a point is analytics or not?

ordinary-differential-equations

So I have the equation

 y" + [(x-x^3)/x]y' +[(sinx)/x]y = 0

My x nought it equal to 0.

I know this is a singular point because my denominator is equal to zero. Then to check if it's analytic or not, I did

 x*P(x) which is equal to (x-x^3)
 x^2*Q(x) which is equal to x(sinx)

How do I determine if this is analytic?


The point $x=0$ is a regular singular point, since

$$ \lim_{x\to 0}xP(x)=\rm{finite}=0, $$

and

$$ \lim_{x\to 0}x^2 Q(x)=\rm{finite}=0 $$

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