Regular singular points (definition)

In summary, the conversation is about understanding the definitions of regular point, regular singular point, and irregular point in relation to a given ODE. The example ODE is x^3y'''(x)+3x^2y''(x)+4xy(x)=0, which after dividing becomes y'''(x)+(3/x)y''(x)+(4/x^2)y(x)=0. It is determined that x=0 is a regular singular point because the order of the pole is lower than the maximum possible. The concept of "poles" comes from complex analysis and in this case, refers to the zeros of the denominator in a quotient of polynomials. It is also mentioned that at a regular singular point, solutions
  • #1
Substance D
49
0
Hello,

I am trying to understand the definition of regular point, regular singular point and irregular point

for example, the ode. what would be the r,rs or i points of this?

x^3y'''(x)+3x^2y''(x)+4xy(x)=0

dividing gives the standard form

y''+(3/x)y' + (4/x^2)y=0

So, obviously x can't equal zero, does that make x a regular singular point because x=0 gives rise to a singularity? If so, what does "regular" mean?

Thanks,
David
 
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  • #2
it seems you divided wrong, or copied the order of the derivatives wrong, or something.

Yes, 0 is a singular point since at least one coefficient has a pole at 0.

In x^3y'''(x)+3x^2y''(x)+4xy(x)=0 it is a regular singular point ... After dividing we have y'''(x)+(3/x)y''(x)+(4/x^2)y(x)=0 and the order of the pole goes up like this: 0,1,0,2 which is lower than the maximum 0,1,2,3 ...

An example irregular singular point: y'''(x)+(3/x^2)y''(x)+(4/x)y(x)=0 now the pole of order 2 in the y'' term is too large.

The reason for this classification is that at a regular singular point the solutions can be written as series in a nice way. At irregular singular points this usually cannot be done.
 
  • #3
Thanks for the reply,

I copied the ODE right, just didn't divide write, good catch though :)

I'm not sure what you mean by "poles".

If something is analytic, it means it can be represented by a series solution, correct?

Thanks,
David
 
  • #4
The idea of "poles" comes from complex analysis. In case of a quotient of polynomials (in lowest terms) the poles are the zeros of the denominator.
 

1. What is the definition of a regular singular point?

A regular singular point is a point in a differential equation where the solution remains bounded and well-behaved as the independent variable approaches the point.

2. How is a regular singular point different from an irregular singular point?

A regular singular point is characterized by having a finite radius of convergence for its power series solution, while an irregular singular point has an infinite radius of convergence.

3. Can a regular singular point be located at the boundary of the domain of a differential equation?

Yes, a regular singular point can occur at the boundary of the domain, as long as the solution remains bounded and well-behaved as the independent variable approaches the point.

4. What is the significance of regular singular points in differential equations?

Regular singular points are important because they allow us to find a power series solution to a differential equation, which can then be used to approximate the exact solution. They are also useful in analyzing the behavior of the solution near the point.

5. Can a regular singular point be converted to a regular point through a change of variables?

Yes, a regular singular point can be transformed into a regular point through a suitable change of variables. This can be useful in solving certain types of differential equations.

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