ODE

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 $r$

Rearrange the equation above

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

Let

Insert this into (*)

At the $x^r$ term, we have

Assuming $a_0\neq 0$,

(**) is called Indicial equation.

There should be 3 cases of roots.

Frobenius theorem: Let $r_1$ and $r_2$ 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, $r_1<r_2$). General solution is given by

 

 

 

 

 

 

Last Updated on 2 years by Yichen Liu