Techniques for Solving Equations with Irregular Singular Points

[stroustroup]

In our differential equations class, we learned about Ordinary and Regular Singular Points of a differential equation. We learned how to solve these equations with power series using the Frobenius method.

I was wondering what happens when there is an irregular singular point, like $x^3y''+2xy'+y=0$ at x = 0.

What kinds of techniques are used to solve these equations (closed-form, series or any other thing besides numerical methods)?

 

[jackmell]

In our differential equations class, we learned about Ordinary and Regular Singular Points of a differential equation. We learned how to solve these equations with power series using the Frobenius method.

I was wondering what happens when there is an irregular singular point, like $x^3y''+2xy'+y=0$ at x = 0.

What kinds of techniques are used to solve these equations (closed-form, series or any other thing besides numerical methods)?

You can start by looking at this thread:

https://www.physicsforums.com/showthread.php?t=711890&highlight=irregular+singular+points

Then pick a simple example with an irregular singular point, solve it, and show the solution has an essential singularity at the singular point. Make some nice (3D) plots too showing the essential singularity (in the complex plane).

 

[AlephZero]

A series solution has no value unless it is convergent. if you know how to get a series solution about the origin, but the origin happens to be a singular point, you can move the origin somewhere else, with a transformation like X = x + a, Y = y + b, or even move it to infinity, with X = 1/x, Y = 1/y.