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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 \(x = x_0\text{.}\) First, we have to solve the “indicial equation”:

\begin{gather*} \lambda (\lambda - 1) + q_0 \lambda + r_0 = 0, \end{gather*}

where

\begin{gather*} q_0 = \lim_{x \to x_0} (x-x_0) \frac{Q(x)}{P(x)} \quad \text{and} \quad r_0 = \lim_{x \to x_0} (x-x_0)^2 \frac{R(x)}{P(x)}. \end{gather*}

The indicial equation has two roots \(\lambda_1\) and \(\lambda_2 \leq \lambda_1\text{.}\) Taking that \(\lambda_1\) is the bigger root, Frobenius' method tells us that the first solution \(y_1\) of the ODE is

\begin{align*} y_1(x) \amp = x^{\lambda_1} \sum_{n=0}^\infty a_n x^n. \end{align*}

The second solution \(y_2\) depends on the second, smaller, root \(\lambda_2\text{.}\)

  1. If \(\lambda_1 - \lambda_2\) is not an integer, then

    \begin{align*} y_2(x) \amp = (x - x_0)^{\lambda_2} \sum_{n=0}^\infty c_n(x-x_0)^n. \end{align*}

  2. If \(\lambda_1 - \lambda_2\) is an integer, then

    \begin{align*} y_2(x) \amp = ay_1(x) \ln |x - x_0| + (x-x_0)^{\lambda_2} \sum_{n=0}^\infty c_n(x - x_0)^n, \end{align*}

    where \(a\) is some real constant. We know for sure that \(a \neq 0\) when \(\lambda_1 = \lambda_2\text{.}\)

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