## 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

Rearrange the equation above

If

Let

Insert this into (*)

At the

Assuming

(**) is called **Indicial equation**.

There should be 3 cases of roots.

**Frobenius theorem**: Let

**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 3 years by Yichen Liu