Frobenius method


Frobenius method

Consider an ODE has the form

The solution can be represented in the form

Bessel’s Equation:

Regular point and singular point

Indicial equation

Rearrange the equation above

If and are analytic, then it can be expanded as power series


Insert this into (*)

At the term, we have

Assuming ,

(**) is called Indicial equation.

There should be 3 cases of roots.

Frobenius theorem: Let and be the roots of the indicial equation, then

  • Case 1: Distinct roots, not differed by integer (e.g. 0, 1/3). General solution is given by

  • Case 2: Double root (e.g. 1/2, 1/2). General solution is given by

  • Case 3: Distinct root, differed by integer (e.g. 0, 1, ). General solution is given by







Last Updated on 2 years by Yichen Liu