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Section 5.1 Definitions

We'll focus on ODEs of the form

\begin{align*} P(x) y'' + Q(x) y' + R(x) y \amp = 0. \end{align*}

Suppose also that \(P(x), Q(x), R(x)\) all have a power series representation around \(x = x_0\text{.}\) This is a very important condition for the next few definitions.

Definition 5.1.1. Ordinary Points.

The point \(x = x_0\) is an ordinary point if \(P(x_0) \neq 0\text{.}\)

Let \(q(x) = Q(x)/P(x)\) and \(r(x) = R(x)/P(x)\text{.}\)

Definition 5.1.2. Singular Points.

The point \(x = x_0\) is a singular point if either \(q(x)\) or \(r(x)\) goes to \(\pm \infty\) as \(x \to x_0\text{.}\)

Definition 5.1.3. Regular Singular Points.

The point \(x = x_0\) is a regular singular point if \(x = x_0\) is a singular point and both limits

\begin{gather*} \lim_{x \to x_0} (x - x_0) q(x) \qquad \text{and} \qquad \lim_{x \to x_0} (x - x_0)^2 r(x) \end{gather*}

are both finite.

Definition 5.1.4. Irregular Singular Points.

If the point \(x = x_0\) is a singular point but is not regular, then the point \(x = x_0\) is an irregular singular point.

Subsection 5.1.1 Why are these defined as they are?

We can solve equations of the form \(x^2 y''' + ax y' + by = 0\) pretty easily when we set \(y = x^r\) (these are Cauchy-Euler ODEs).

Now, we want to solve equations of the form \(P(x) y'' + Q(x) y' + R(x) y = 0\text{.}\) By dividing out by \(P(x)\text{,}\) we can transform this into the equation \(y'' + q(x) y' + r(x)y = 0\) where

\begin{gather*} q(x) = \frac{Q(x)}{P(x)} \qquad \text{and} \qquad r(x) = \frac{R(x)}{P(x)}. \end{gather*}

To try to kinda match the Cauchy-Euler ODE form, multiply both sides of \(y'' + q(x) y' + r(x) y = 0\) by \(x^2\) to get

\begin{gather*} x^2 y'' + x^2 q(x) y' + x^2 r(x) y = 0. \end{gather*}

Then, to further match the form of the constants in the Cauchy-Euler ODEs, group the coefficients like this:

\begin{gather*} x^2 y'' + [x q(x)] xy' + [x^2r(x)] y = 0. \end{gather*}

This way, if \(xq(x) = a\) and \(x^2r(x) = b\text{,}\) then we'd easily get back to a Cauchy-Euler ODE. More realistically, you'd want \(xq(x)\) and \(x^2r(x)\) to be representable as a power series.

For example, say we set \(xq(x)\) equal to

\begin{align*} xq(x) \amp = q_0 + q_1 x + q_2 x^2 + \cdots + q_nx^n + \cdots. \end{align*}

Then, when we evaluate \(xq(x)\) at \(x = 0\text{,}\) we get the importantly finite value \(q_0\text{.}\) This corresponds to saying that the limit

\begin{gather*} \lim_{x \to 0} xq(x) \end{gather*}

is finite and equal to \(q_0\text{.}\) A similar thing happens with \(x^2r(x)\text{.}\)