Math292 Textbook Frobenius' Method

Section 7.1 Frobenius' Method

Our previous method worked for ordinary points, but the same technique won't exactly work for regular singular points.

Say that we want to solve the ODE

P (x) y^{″} + Q (x) y^{'} + R (x) y = 0

around the regular singular point

x = x_{0} .

First, we have to solve the “indicial equation”:

\begin{matrix} λ (λ - 1) + q_{0} λ + r_{0} = 0, \end{matrix}

where

\begin{matrix} q_{0} = lim_{x \to x_{0}} (x - x_{0}) \frac{Q (x)}{P (x)} and r_{0} = lim_{x \to x_{0}} (x - x_{0})^{2} \frac{R (x)}{P (x)} . \end{matrix}

The indicial equation has two roots

λ_{1}

and

λ_{2} \leq λ_{1} .

Taking that

λ_{1}

is the bigger root, Frobenius' method tells us that the first solution

y_{1}

of the ODE is

\begin{aligned} y_{1} (x) & = x^{λ_{1}} \sum_{n = 0}^{\infty} a_{n} x^{n} . \end{aligned}

The second solution

y_{2}

depends on the second, smaller, root

λ_{2} .

If $λ_{1} - λ_{2}$ is not an integer, then

$\begin{aligned} y_{2} (x) & = (x - x_{0})^{λ_{2}} \sum_{n = 0}^{\infty} c_{n} (x - x_{0})^{n} . \end{aligned}$
If $λ_{1} - λ_{2}$ is an integer, then

$\begin{aligned} y_{2} (x) & = a y_{1} (x) \ln | x - x_{0} | + (x - x_{0})^{λ_{2}} \sum_{n = 0}^{\infty} c_{n} (x - x_{0})^{n}, \end{aligned}$

where $a$ is some real constant. We know for sure that $a \neq 0$ when $λ_{1} = λ_{2} .$

Often times, we'll only need to know the form of these solutions, not necessarily the specifics.