Isn't Ordinary Point Just a Specific Case of Regular Singular Point?

In summary, if the limits of P(x) and Q(x) converge at x_o, it can also be considered a regular singular point for a second-order linear ODE. However, this does not mean that an ordinary point can be classified as a regular singular point or an irregular singular point. The classification of a point is dependent on its own criteria.
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gikiian
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Consider the ODE [itex] [/itex] [itex]y''+P(x)y'+Q(x)y=0[/itex].
If [itex]\stackrel{limit}{_{x→x_{o}}}P(x)[/itex] and [itex]\stackrel{limit}{_{x→x_{o}}}Q(x)[/itex] converge, can you call [itex]x_{o}[/itex] a 'regular singular point' besides calling it an 'ordinary point'?

I am saying this because if [itex]\stackrel{limit}{_{x→x_{o}}}P(x)[/itex] and [itex]\stackrel{limit}{_{x→x_{o}}}Q(x)[/itex] converge, then [itex]\stackrel{limit}{_{x→x_{o}}}(x-x_{o})P(x)[/itex] and [itex]\stackrel{limit}{_{x→x_{o}}}(x-x_{o})^{2}Q(x)[/itex] will also converge. And for a second-order linear ODE for which [itex]\stackrel{limit}{_{x→x_{o}}}(x-x_{o})P(x)[/itex] and [itex]\stackrel{limit}{_{x→x_{o}}}(x-x_{o})^{2}Q(x)[/itex] converge, [itex]x_{o}[/itex] is termed as a regular singular point.
 
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  • #2
I see what you mean. An ordinary point satisfies the requirements of a regular singular point. For most purposes we want to classify the point (ie ordinary, regular singular, or irregular singular). Thus we would not call a regular singular point an irregular singular point nor an ordinary point a regular singular point.
 
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  • #3
I totally get that! Thanks :)
 

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

An ordinary point is a point in a differential equation where the functions involved are well-behaved and can be expressed as a Taylor series. On the other hand, a regular singular point is a point where one or more of the functions involved in the differential equation have singularities, making them difficult to express as a Taylor series.

2. Can an ordinary point also be a regular singular point?

Yes, it is possible for a point to be both an ordinary point and a regular singular point. This occurs when the functions involved have a singularity at that point, but the singularity can be "removed" by a change of variables.

3. How does knowing the type of point affect the solution of a differential equation?

Knowing the type of point can help determine the type of solution that the differential equation will have. For ordinary points, the solution will typically involve a power series. For regular singular points, the solution may involve special functions such as Bessel functions or hypergeometric functions.

4. Are there any other types of points in differential equations?

Yes, there are other types of points in differential equations, such as irregular singular points and essential singular points. These points have more complicated behavior and can result in more complex solutions.

5. Can the type of point change depending on the form of the differential equation?

Yes, the type of point can change depending on the form of the differential equation. For example, a regular singular point in the form of a second-order differential equation may become an ordinary point when the equation is written in a different form. Therefore, it is important to consider the form of the equation when determining the type of point.

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