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Singularities classification in DE's

  1. Jan 31, 2012 #1

    fluidistic

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    Gold Member

    According to Mathworld, if in [itex]y''+P(x)y'+Q(x)y=0[/itex], P diverges at [itex]x=x_0[/itex] quicker than [itex]\frac{1}{(x-x_0)}[/itex] or Q diverges at [itex]x=x_0[/itex] quicker than [itex]\frac{1}{(x-x_0)^2}[/itex] then [itex]x_0[/itex] is called an essential singularity.
    What I don't understand is that let's suppose Q diverges like [itex]\frac{1}{(x-x_0)^5}[/itex]. 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?
     
  2. jcsd
  3. Jan 31, 2012 #2
    If [itex]x_0[/itex] is a pole of order k for the function, then it should behave as:
    [tex]
    y(x) \sim \frac{1}{(x - x_0)^k}
    [/tex]
    then:
    [tex]
    y'(x) \sim \frac{-k}{(x - x_0)^{k + 1}}
    [/tex]
    and
    [tex]
    y''(x) \sim \frac{k (k + 1)}{(x - x_0)^{k + 2}}
    [/tex]
    How do you propose to make an equality using:
    [tex]
    Q(x) \, y(x) \sim \frac{1}{(x - x_0)^{k + 5}}
    [/tex]
    ?
     
  4. Jan 31, 2012 #3

    fluidistic

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    Gold Member

    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 [itex]\frac{1}{(x-x_0)^5}[/itex]. It's an essential singularity because it diverges quicker than [itex]\frac{1}{(x-x_0)^2}[/itex] when [itex]x[/itex] tends to [itex]x_0[/itex]."
    But if I use the definition of a pole of order n for a function, namely that [itex]\lim _{x\to x_0} (x-x_0)^nf(x)[/itex] is differentiable at [itex]x=x_0[/itex], where n is the smallest integer and where [itex]f(x)=Q(x)[/itex], I get that [itex]\lim _{x\to x_0} (x-x_0)^5Q(x)=1[/itex] which is clearly differentiable at [itex]x=x_0[/itex]. For n=4, it isn't differentiable in [itex]x=x_0[/itex].
    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!
     
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