Singular Points of the ODE: Identifying and Understanding

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Homework Help Overview

The discussion revolves around identifying and understanding singular points of the ordinary differential equation (ODE) xy" + (2-x)y' + y = 0. Participants are exploring the nature of singular points in the context of the equation's coefficients.

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants discuss the method of finding p(x) and q(x) to determine analyticity, with some questioning the meaning of "SF" and the definitions of ordinary and singular points. There is an exploration of how to show the equation has only one singular point.

Discussion Status

The discussion is ongoing, with participants seeking clarification on definitions and methods. Some guidance has been offered regarding the identification of p(x) and q(x), but there is no consensus on the nature of the singular point or how to demonstrate its uniqueness.

Contextual Notes

There is a lack of clarity regarding the definitions of terms such as "SF" and the specific conditions under which the coefficients are considered analytic. Participants are also navigating the implications of dividing by x in the context of the ODE.

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Homework Statement


For the ODE xy" + (2-x)y' + y = 0

i want to show it has one singular point and identify its nature


Homework Equations





The Attempt at a Solution



I have read the topic and I see that a point Xo is called and ordinary point of the equation if both p(x) and q(x) (once converted to SF) are anlytic at Xo.

I really don't understand the method to work this out though...

Help would be much appreciated
 
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What "method" are you talking about? Just finding p(x) and q(x) and determining whether they are analytic?

Unfortunately, you haven't said what you mean by "SF" nor what p(x) and q(x) are here.

I might guess that by "SF" (standard form?) You mean the equation solved for y". Here that would be y"= ((2-x)/x) y'+ (1/x)y and perhaps you mean p(x)= (2-x)/x and q(x)= 1/x.
 
yes i am aware that when we divide by x this ODE is then in standard form. What i don't understand is how to show this equation has only one singular point?
 
What IS a singular point?

You've already said that an "ordinary point" is a point where the coefficents of y' and y are not analytic. A "singular point" is a point where that is not true. For what values of x and y are (2-x)/x and 1/x not analytic?
 

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