Singular Points of the ODE: Identifying and Understanding

In summary, the conversation discusses an ODE and the desire to show that it has one singular point and determine its nature. The method to do so is not fully understood and help is requested. The terms "SF" (standard form) and "ordinary point" are defined and the concept of a singular point is briefly explained.
  • #1
cabellos6
31
0

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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  • #2
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.
 
  • #3
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?
 
  • #4
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?
 

What is a singular point of an ODE?

A singular point of an ODE (ordinary differential equation) is a point where the solution of the equation becomes infinite or undefined. It can also be described as a point where the behavior of the solution changes significantly.

How do you identify a singular point of an ODE?

A singular point can be identified by setting the coefficients of the highest derivative term in the equation to zero and solving for the independent variable. The resulting values of the independent variable are the singular points of the ODE.

What is the significance of singular points in solving ODEs?

Singular points are important because they can affect the behavior and stability of the solution of an ODE. They can also indicate the presence of special solutions, such as periodic or oscillating solutions.

Can an ODE have multiple singular points?

Yes, an ODE can have multiple singular points. These can be classified as regular or irregular singular points, depending on the behavior of the solution at that point.

How are singular points classified?

Singular points can be classified based on the type of singularity they exhibit. Regular singular points have a solution that can be expressed as a power series, while irregular singular points have a solution that cannot be expressed as a power series.

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