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

Click For Summary
SUMMARY

The discussion clarifies the classification of points in the context of second-order linear ordinary differential equations (ODEs) represented by the equation y'' + P(x)y' + Q(x)y = 0. It establishes that if the limits of P(x) and Q(x) converge as x approaches x₀, then x₀ can be classified as a regular singular point. However, while an ordinary point meets the criteria for a regular singular point, it is essential to maintain distinct classifications for ordinary, regular singular, and irregular singular points to avoid confusion.

PREREQUISITES
  • Understanding of second-order linear ordinary differential equations (ODEs)
  • Familiarity with the concepts of ordinary points and singular points
  • Knowledge of limits and convergence in mathematical analysis
  • Basic proficiency in differential equations and their classifications
NEXT STEPS
  • Study the classification of singular points in differential equations
  • Learn about the Frobenius method for solving ODEs with singular points
  • Explore the implications of regular singular points in the context of solutions
  • Investigate the differences between ordinary points and irregular singular points
USEFUL FOR

Mathematicians, students of differential equations, and educators seeking to deepen their understanding of point classifications in ODEs will benefit from this discussion.

gikiian
Messages
98
Reaction score
0
Consider the ODE y''+P(x)y'+Q(x)y=0.
If \stackrel{limit}{_{x→x_{o}}}P(x) and \stackrel{limit}{_{x→x_{o}}}Q(x) converge, can you call x_{o} a 'regular singular point' besides calling it an 'ordinary point'?

I am saying this because if \stackrel{limit}{_{x→x_{o}}}P(x) and \stackrel{limit}{_{x→x_{o}}}Q(x) converge, then \stackrel{limit}{_{x→x_{o}}}(x-x_{o})P(x) and \stackrel{limit}{_{x→x_{o}}}(x-x_{o})^{2}Q(x) will also converge. And for a second-order linear ODE for which \stackrel{limit}{_{x→x_{o}}}(x-x_{o})P(x) and \stackrel{limit}{_{x→x_{o}}}(x-x_{o})^{2}Q(x) converge, x_{o} is termed as a regular singular point.
 
Last edited:
Physics news on Phys.org
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.
 
  • Like
Likes   Reactions: 1 person
I totally get that! Thanks :)
 

Similar threads

Graduate Few questions about series solution of ODEs
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Runge-Kutta 4 w/ some sugar on the top: How to do error approximation?
  • · Replies 65 ·
3
Replies
65
Views
9K
Graduate Ordinary points, regular singular points and irregular singular points
  • · Replies 3 ·
Replies
3
Views
20K
Undergrad Correct Upper/Lower limits for continuation of solutions for ODE
  • · Replies 0 ·
Replies
0
Views
4K
MHB Fixed point iteration: g is a contraction mapping
  • · Replies 17 ·
Replies
17
Views
4K
Undergrad Converting Second Order ODE to Hypergeometric Function
  • · Replies 7 ·
Replies
7
Views
3K
Graduate Green's function for Sturm-Louiville ODE
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad 2nd order ODE's, variation of parameters and the notorious constraint
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Revisiting the Light Clock
  • · Replies 25 ·
Replies
25
Views
2K
Graduate 2D Stationary advection-diffusion eq. as a BVP
  • · Replies 6 ·
Replies
6
Views
2K