Math292 Textbook Definitions

Section 5.1 Definitions

We'll focus on ODEs of the form

\begin{aligned} P (x) y^{″} + Q (x) y^{'} + R (x) y & = 0. \end{aligned}

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Suppose also that

P (x), Q (x), R (x)

all have a power series representation around

x = x_{0} .

This is a very important condition for the next few definitions.

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Definition 5.1.1. Ordinary Points.

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

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Let

q (x) = Q (x) / P (x)

and

r (x) = R (x) / P (x) .

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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 $x \to x_{0} .$

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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{matrix} lim_{x \to x_{0}} (x - x_{0}) q (x) and lim_{x \to x_{0}} (x - x_{0})^{2} r (x) \end{matrix}

are both finite.

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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.

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Subsection 5.1.1 Why are these defined as they are?

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We can solve equations of the form

x^{2} y^{‴} + a x y^{'} + b y = 0

pretty easily when we set

y = x^{r}

(these are Cauchy-Euler ODEs).

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Now, we want to solve equations of the form

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

By dividing out by

P (x),

we can transform this into the equation

y^{″} + q (x) y^{'} + r (x) y = 0

where

\begin{matrix} q (x) = \frac{Q (x)}{P (x)} and r (x) = \frac{R (x)}{P (x)} . \end{matrix}

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To try to kinda match the Cauchy-Euler ODE form, multiply both sides of

y^{″} + q (x) y^{'} + r (x) y = 0

x^{2}

to get

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

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

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

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This way, if

x q (x) = a

and

x^{2} r (x) = b,

then we'd easily get back to a Cauchy-Euler ODE. More realistically, you'd want

x q (x)

and

x^{2} r (x)

to be representable as a power series.

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For example, say we set

x q (x)

equal to

\begin{aligned} x q (x) & = q_{0} + q_{1} x + q_{2} x^{2} + \dots + q_{n} x^{n} + \dots . \end{aligned}

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Then, when we evaluate

x q (x)

x = 0,

we get the importantly finite value

q_{0} .

This corresponds to saying that the limit

\begin{matrix} lim_{x \to 0} x q (x) \end{matrix}

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is finite and equal to

q_{0} .

A similar thing happens with

x^{2} r (x) .