Singularities classification in DE's

In summary, an essential singularity is defined as a point where the coefficients of the highest order derivatives diverge faster than the inverse of the distance from that point. This is in contrast to a pole, which is defined as a point where the function itself diverges faster than the inverse of the distance from that point. This distinction can be seen in the example of Q(x) behaving like \frac{1}{(x-x_0)^5}, which would be considered an essential singularity due to the divergence of its coefficients, even though it may appear to be a pole of order 5.
  • #1
fluidistic
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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?
 
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  • #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]
?
 
  • #3
Dickfore said:
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]
?
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!
 

1. What is a singularity in differential equations (DEs)?

A singularity in DEs refers to a point in a function where the function becomes undefined or infinite. It can occur when the solution to a DE approaches zero or infinity, or when a point on the function has a vertical tangent.

2. How are singularities classified in DEs?

Singularities in DEs can be classified as either removable or essential. A removable singularity is one where the function can be modified or extended to remove the singularity. An essential singularity, on the other hand, cannot be removed or extended, and the function will remain undefined at that point.

3. What is a pole in DEs?

A pole in DEs is a specific type of essential singularity. It occurs when the function approaches infinity at a point, and the singularity is characterized by the order of the pole. A pole of order n means that the function becomes infinite at that point with a rate of n.

4. How do singularities affect the behavior of a DE solution?

Singularities can greatly impact the behavior of a DE solution. A removable singularity may not have a significant effect, but an essential singularity can cause the solution to behave unpredictably or have a discontinuity. In some cases, singularities can also provide valuable information about the nature of the DE.

5. Can singularities be avoided or eliminated in DEs?

Singularities cannot be avoided in all cases, but in some situations, they can be eliminated or moved to a different location by transforming the DE. However, eliminating a singularity may also change the nature of the DE and the solution. In general, singularities are an inherent part of DEs and cannot be completely avoided.

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