8.6 Method of Frobenius
Regular and Irregular Singular Points A singular point π₯0 of
π¦" + π(π₯)π¦β² + π(π₯)π¦ = 0
Is said to be a regular singular point if the functions
π(π₯) = (π₯ β π₯0)π(π₯)
π(π₯) = (π₯ β π₯0)
2π(π₯)
Are both analytic at π₯0. Otherwise, π₯0 is called an irregular singular point.
Example
1. Determine the singular points of the differential equation. Classify them as regular or irregular.
π₯(π₯ + 3)3π¦β²β² β π¦ = 0
Frobeniusβ Theorem
If π₯0 is a regular singular point of the differential equation
π2(π₯)π¦" + π1(π₯)π¦β² + π0(π₯)π¦ = 0
Then there exists at least one series solution of the form
π¦ = βππ(π₯ β π₯0)π+π
β
π=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 < π₯ β π₯0 < π , where R is the distance from π₯0 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
π¦ = βππ(π₯ β π₯0)π+π
β
π=0
Now, in addition to finding ππ we will also have to find r.
Note: As in previous sections we will look at examples with the regular singular point π₯0 = 0.
Indicial Equation If π₯0 is a regular singular point of
π¦" + π(π₯)π¦β² + π(π₯)π¦ = 0
Then the indicial equation for this point is
π(π β 1) + π0π + π0 = 0
where
π0 = limπ₯βπ₯0
(π₯ β π₯0)π(π₯)
π0 = lim
π₯βπ₯0(π₯ β π₯0)
2π(π₯)
The roots of the indicial equation are called the exponents (indices) of the singularity π₯0.
Examples
2. Find the indicial equation and the exponents at the singularity π₯ = 0.
π₯2π¦β²β² + 4π₯π¦β² + 2π¦ = 0
3. Find a series solution about the regular singular point π₯ = 0 of
2π₯2π¦β²β² β π₯π¦β² + (1 + π₯)π¦ = 0
4. Find a series solution about the regular singular point π₯ = 0 of
2π₯2π¦β²β² β π₯π¦β² + (π₯2 + 1)π¦ = 0
Top Related