# Classifying ordinary and singular points

• Odyssey
In summary, the first example has an ordinary point at x=0 because (sin x)/x is infinitely differentiable, while the second example has a singular point at x=0 due to the coefficient of y'' approaching 0 as x approaches 0.
Odyssey
Hello,

I am stuck on classifying the points with this DE...=\
xy''+(x-x^3)y'+(sin x)y=0
The solution says (sin x)/x is infinitely differentiable...so x=0 is an ordinary point?

I was taught...if P(xo)=0, then xo is a singular point. Here P(x)=x...so x=0. So, what I don't get is the "infintely differentiable" part. Does it have something to do with the convergent Tayloe Series about x=0? And what does that mean?

I got another example here...
x^2(y'')+(cos x)y'+xy=0
Here P(x)=0 so x=0 is a singular point. It's also a irregular singular point because as x->0 (cos x)/x goes to infinity. How come here x=0 is a singular point here?

Thank you.

In the first example, the fact that (sin x)/x is infinitely differentiable means that the differential equation can be written as a regular one near x=0, so it is an ordinary point. In the second example, x=0 is a singular point because x^2 in the coefficient of y'' goes to 0 as x approaches 0, which makes the DE have a removable singularity at x=0. An irregular singular point refers to a point where the coefficients of the highest order derivatives blow up or go to zero as x approaches the point.

## 1. What is the difference between an ordinary point and a singular point?

An ordinary point is a point on a function where the function is defined and continuous. A singular point, on the other hand, is a point where the function is not defined or is discontinuous.

## 2. How do you classify a point as ordinary or singular?

To classify a point, you need to analyze the behavior of the function at that point. If the function is continuous and defined at that point, it is an ordinary point. If the function is not defined or discontinuous at that point, it is a singular point.

## 3. Can an ordinary point become a singular point?

Yes, an ordinary point can become a singular point if the function becomes discontinuous at that point. This can happen when the function has a vertical asymptote or a removable discontinuity at that point.

## 4. Why is it important to classify ordinary and singular points?

Classifying ordinary and singular points helps us understand the behavior of a function and identify any potential issues such as discontinuities. This information is crucial in solving differential equations and understanding the overall behavior of a function.

## 5. What are the applications of classifying ordinary and singular points?

Classifying ordinary and singular points is important in many areas of science and engineering, such as physics, chemistry, and electrical engineering. It is used in solving differential equations, analyzing the stability of systems, and understanding the behavior of physical systems.

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