8.6 Method of Frobenius

Regular and Irregular Singular Points A singular point 𝑥0 of

𝑦" + 𝑝(𝑥)𝑦′ + 𝑞(𝑥)𝑦 = 0

Is said to be a regular singular point if the functions

𝑃(𝑥) = (𝑥 − 𝑥0)𝑝(𝑥)

𝑄(𝑥) = (𝑥 − 𝑥0)

2𝑞(𝑥)

Are both analytic at 𝑥0. Otherwise, 𝑥0 is called an irregular singular point.

Example

1. Determine the singular points of the differential equation. Classify them as regular or irregular.

𝑥(𝑥 + 3)3𝑦′′ − 𝑦 = 0

Frobenius’ Theorem

If 𝑥0 is a regular singular point of the differential equation

𝑎2(𝑥)𝑦" + 𝑎1(𝑥)𝑦′ + 𝑎0(𝑥)𝑦 = 0

Then there exists at least one series solution of the form

𝑦 = ∑𝑐𝑛(𝑥 − 𝑥0)𝑛+𝑟

∞

𝑛=0

Where the number r is a constant referred to as the indicial root or exponent..

The series converges at least on some interval 0 < 𝑥 − 𝑥0 < 𝑅, where R is the distance from 𝑥0 to the nearest

other singular point of the differential equation.

The Method of Frobenius

This method is similar to the method of 8.3, only instead of using a power series for y we will use

𝑦 = ∑𝑐𝑛(𝑥 − 𝑥0)𝑛+𝑟

∞

𝑛=0

Now, in addition to finding 𝑎𝑛 we will also have to find r.

Note: As in previous sections we will look at examples with the regular singular point 𝑥0 = 0.

Indicial Equation If 𝑥0 is a regular singular point of

𝑦" + 𝑝(𝑥)𝑦′ + 𝑞(𝑥)𝑦 = 0

Then the indicial equation for this point is

𝑟(𝑟 − 1) + 𝑝0𝑟 + 𝑞0 = 0

where

𝑝0 = lim𝑥→𝑥0

(𝑥 − 𝑥0)𝑝(𝑥)

𝑞0 = lim

𝑥→𝑥0(𝑥 − 𝑥0)

2𝑞(𝑥)

The roots of the indicial equation are called the exponents (indices) of the singularity 𝑥0.

Examples

2. Find the indicial equation and the exponents at the singularity 𝑥 = 0.

𝑥2𝑦′′ + 4𝑥𝑦′ + 2𝑦 = 0

3. Find a series solution about the regular singular point 𝑥 = 0 of

2𝑥2𝑦′′ − 𝑥𝑦′ + (1 + 𝑥)𝑦 = 0

4. Find a series solution about the regular singular point 𝑥 = 0 of

2𝑥2𝑦′′ − 𝑥𝑦′ + (𝑥2 + 1)𝑦 = 0