Singularities classification in DE's

[fluidistic]

According to Mathworld, if in y''+P(x)y'+Q(x)y=0, P diverges at x=x_0 quicker than \frac{1}{(x-x_0)} or Q diverges at x=x_0 quicker than \frac{1}{(x-x_0)^2} then x_0 is called an essential singularity.
What I don't understand is that let's suppose Q diverges like \frac{1}{(x-x_0)^5}. In that case x_0 would be called an essential singularity. But what I don't understand is that to me it looks like a pole of order 5, not an essential singularity (pole of order infinity).
Am I missing something?

 

[Dickfore]

If x_0 is a pole of order k for the function, then it should behave as:
<br /> y(x) \sim \frac{1}{(x - x_0)^k}<br />
then:
<br /> y&#039;(x) \sim \frac{-k}{(x - x_0)^{k + 1}}<br />
and
<br /> y&#039;&#039;(x) \sim \frac{k (k + 1)}{(x - x_0)^{k + 2}}<br />
How do you propose to make an equality using:
<br /> Q(x) \, y(x) \sim \frac{1}{(x - x_0)^{k + 5}}<br />
?

 

[fluidistic]

If x_0 is a pole of order k for the function, then it should behave as:
<br /> y(x) \sim \frac{1}{(x - x_0)^k}<br />
then:
<br /> y&#039;(x) \sim \frac{-k}{(x - x_0)^{k + 1}}<br />
and
<br /> y&#039;&#039;(x) \sim \frac{k (k + 1)}{(x - x_0)^{k + 2}}<br />
How do you propose to make an equality using:
<br /> Q(x) \, y(x) \sim \frac{1}{(x - x_0)^{k + 5}}<br />
?

Hmm I don't really understand your question. We're talking about a pole/singularity for Q or P right? Not y(x)... or I'm wrong on this?
So it would be "let's say Q(x) behaves like \frac{1}{(x-x_0)^5}. It's an essential singularity because it diverges quicker than \frac{1}{(x-x_0)^2} when x tends to x_0."
But if I use the definition of a pole of order n for a function, namely that \lim _{x\to x_0} (x-x_0)^nf(x) is differentiable at x=x_0, where n is the smallest integer and where f(x)=Q(x), I get that \lim _{x\to x_0} (x-x_0)^5Q(x)=1 which is clearly differentiable at x=x_0. For n=4, it isn't differentiable in x=x_0.
I know I'm missing something but I still don't see it. Could you be more specific please?
Thank you so far for your answer!