Regular singular points of 2nd order ODE

In summary, the conversation discusses finding the ansatz for q in a 2nd order ODE with regular singular points at z=a and z=b. The most general form for q is given as q(x)=\frac{g(z)}{(z-a)^2(z-b)^2}, where g(z) is analytic everywhere. The linearity condition is also mentioned as a method for finding the functions f and g.
  • #1
Jerbearrrrrr
127
0

Homework Statement


[PLAIN]http://img265.imageshack.us/img265/6778/complex.png

I did the coefficient of the w' term. What about the w term?

This seems like a fairly standard thing, but I can't seem to find it anywhere.
What ansatz should I use for q, if the eqn is written w''+pw'+qw?
C/(z-a)²+ D/(z-b)²?
Any conditions, except for the one generated by z->1/t substitution?
Or should I use C+cz on the top, etc?
 
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  • #2
Jerbearrrrrr said:

Homework Statement


[PLAIN]http://img265.imageshack.us/img265/6778/complex.png

I did the coefficient of the w' term. What about the w term?

This seems like a fairly standard thing, but I can't seem to find it anywhere.
What ansatz should I use for q, if the eqn is written w''+pw'+qw?
C/(z-a)²+ D/(z-b)²?
Any conditions, except for the one generated by z->1/t substitution?
Or should I use C+cz on the top, etc?

Well, if a 2nd order ODE has regular singular points at [itex]z=a[/itex] and [itex]z=b[/itex], then [itex]q[/itex] has poles up to 2nd order at those points, and the most general form of [itex]q[/itex] is then

[tex]q(x)=\frac{g(z)}{(z-a)^2(z-b)^2}[/tex]

where [itex]g(z)[/itex] is analytic everywhere. You should have used similar reasoning to find

[tex]p(z)=\frac{f(z)}{(z-a)(z-b)}[/itex]

Then just apply the linearity condition to find [itex]f[/itex] and [itex]g[/itex].
 
Last edited by a moderator:

FAQ: Regular singular points of 2nd order ODE

1. What is a regular singular point in a second order ODE?

A regular singular point in a second order ordinary differential equation (ODE) is a point where the coefficient of the highest order derivative term is zero. This causes the ODE to have a singularity at that point, meaning that the solution may become unbounded or undefined. However, the singularity is "regular" because the other coefficients in the equation are well-behaved and do not cause any additional issues.

2. How do you identify regular singular points in a second order ODE?

To identify regular singular points in a second order ODE, you can examine the coefficients of the highest order derivative term and see if any of them are zero. If so, those points are considered regular singular points. You can also plot the equation to visually identify any points where the solution becomes unbounded or undefined.

3. What types of differential equations have regular singular points?

Regular singular points can occur in different types of differential equations, such as linear and non-linear equations. However, they are most commonly found in linear differential equations with variable coefficients. These equations often arise in physical and engineering problems.

4. Can regular singular points be avoided in second order ODEs?

In some cases, regular singular points can be avoided by transforming the ODE into a different form. This is often done by using a change of variables or by making a substitution. However, in some cases, it may not be possible to avoid regular singular points, and they must be dealt with using special techniques.

5. How are regular singular points handled in solving second order ODEs?

Regular singular points require special techniques to be handled in solving second order ODEs. One common method is to use the Frobenius method, which involves finding a power series solution to the equation. Another approach is to use a transformation to reduce the equation to a simpler form without the regular singular point. In some cases, numerical methods may also be used to approximate the solution near the singular point.

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