Power Series Solutions

Power Series Solutions

Motivation

Ordinary Points

Power Series Solution

At an ordinary point , the general solution is:

Method

1. Assume
2. Compute and
3. Substitute into the ODE
4. Collect like powers of
5. Set coefficients to zero to find recurrence relation
6. Solve for in terms of and

Example

,

Substituting:

Recurrence: for ,

Singular Points

Frobenius Method

Indicial Equation

Substituting the Frobenius series gives an indicial equation for :

Three Cases

Bessel's Equation

Solutions are Bessel functions and .

Legendre's Equation

Solutions are Legendre polynomials .

Radius of Convergence

Applications