Question

The definitions of an ordinary point and a regular singular point given in the preceding sections apply only if the point \(x_{0}\) is finite. In more advanced work in differential equations it is often necessary to discuss the point at infinity. This is done by making the change of variable \(\xi=1 / x\) and studying the resulting equation at \(\xi=0 .\) Show that for the differential equation \(P(x) y^{\prime \prime}+Q(x) y^{\prime}+R(x) y=0\) the point at infinity is an ordinary point if $$ \frac{1}{P(1 / \xi)}\left[\frac{2 P(1 / \xi)}{\xi}-\frac{Q(1 / \xi)}{\xi^{2}}\right] \quad \text { and } \quad \frac{R(1 / \xi)}{\xi^{4} P(1 / \xi)} $$ have Taylor series expansions about \(\xi=0 .\) Show also that the point at infinity is a regular singular point if at least one of the above functions does not have a Taylor series expansion, but both \(\frac{\xi}{P(1 / \xi)}\left[\frac{2 P(1 / \xi)}{\xi}-\frac{Q(1 / \xi)}{\xi^{2}}\right] \quad\) and \(\quad \frac{R(1 / \xi)}{\xi^{2} P(1 / \xi)}\) do have such expansions.

Short answer

Answer: In order to determine if the point at infinity is an ordinary point or a regular singular point, we must analyze the functions mentioned in the step-by-step solution for the given differential equation and see if they satisfy the specified conditions.

Step-by-step solution

Change of Variable

First, we should change the variable from \(x\) to \(\xi\) using the relation \(\xi = \frac{1}{x}\). This implies that \(x=\frac{1}{\xi}\). We'll also need the derivatives \(y'(x)\) and \(y''(x)\) in terms of \(\xi\): 1. To find \(y'(x)\) in terms of \(\xi\), first observe that \(\frac{dx}{d\xi} = -\frac{1}{\xi^2}\). So, we get \(\frac{dy}{dx} = \frac{dy}{d\xi} \frac{d\xi}{dx} = -\xi^2 \frac{dy}{d\xi}\). 2. To find \(y''(x)\) in terms of \(\xi\), differentiate \(y'(x)\) with respect to \(x\): \(\frac{d^2y}{dx^2} = \frac{d}{dx} (-\xi^2 \frac{dy}{d\xi})\). Then, applying the chain rule we get: \(\frac{d^2y}{dx^2} = \frac{d}{d\xi} (-\xi^2 \frac{dy}{d\xi}) \frac{d\xi}{dx} = -\frac{1}{\xi^2}\frac{d}{d\xi}(-\xi^2 \frac{dy}{d\xi}) = \xi^4\frac{d^2y}{d\xi^2}+2\xi^3 \frac{dy}{d\xi}\).

Transform the Differential Equation

Replace \(y'(x)\) and \(y''(x)\) in terms of \(\xi\) in the original differential equation: \(P(x)y''(x) + Q(x)y'(x) + R(x)y=0\). Substitute \(x=\frac{1}{\xi}\) and the expressions for \(y'(x)\) and \(y''(x)\) from step 1: $$P\left(\frac{1}{\xi}\right)(\xi^4\frac{d^2y}{d\xi^2}+2\xi^3 \frac{dy}{d\xi}) + Q\left(\frac{1}{\xi}\right) (-\xi^2 \frac{dy}{d\xi}) + R\left(\frac{1}{\xi}\right)y = 0$$

Conditions for an Ordinary Point

We will check if the point at infinity is an ordinary point by finding whether the given quantities have Taylor expansions about \(\xi=0\): 1. \(\frac{1}{P(1 / \xi)}\left[\frac{2 P(1 / \xi)}{\xi}-\frac{Q(1 /\xi)}{\xi^{2}}\right]\) 2. \(\frac{R(1 / \xi)}{\xi^{4} P(1 / \xi)}\) If both of them have Taylor series expansions about \(\xi=0\), the point at infinity is an ordinary point.

Conditions for a Regular Singular Point

Now, let's check if the point at infinity is a regular singular point. If at least one of the above functions in step 3 does not have a Taylor series expansion, but both of the following functions have Taylor series expansions about \(\xi=0\), the point at infinity is a regular singular point: 1. \(\frac{\xi}{P(1 / \xi)}\left[\frac{2 P(1 / \xi)}{\xi}-\frac{Q(1 /\xi)}{\xi^{2}}\right]\) 2. \(\frac{R(1 / \xi)}{\xi^{2} P(1 / \xi)}\) In order to determine if infinity is an ordinary or regular singular point, we must inspect the functions mentioned in steps 3 and 4 for the given differential equation and see if they satisfy the described conditions.

Key concepts

Ordinary Point

In differential equations, an ordinary point for a given differential equation is a point where the functions involved behave in a regular, predictable manner. The notion is centered around the presence of Taylor series expansions of the given functions around that point.

For a differential equation like \(P(x) y'' + Q(x) y' + R(x) y = 0\), if changing variables to \(\xi = \frac{1}{x}\), then studying the point at infinity translates to analyzing around \(\xi = 0\). If both functions:

• \( \frac{1}{P(1 / \xi)}\left[\frac{2 P(1 / \xi)}{\xi} - \frac{Q(1 / \xi)}{\xi^{2}}\right] \)
• \( \frac{R(1 / \xi)}{\xi^{4} P(1 / \xi)} \)

have Taylor series expansions about \(\xi = 0\), infinity is considered an ordinary point.

This implies that the behavior of the differential equation remains smooth and well-defined, which simplifies the solution process. Recognizing ordinary points helps in anticipating whether solutions can be expressed in power series or other well-understood functional forms.

Regular Singular Point

A regular singular point arises at locations where the solution to a differential equation may have a singularity, but still remains manageable under specific conditions. Consider the differential equation given as:
\(P(x) y'' + Q(x) y' + R(x) y = 0\).
When variable substitution \(\xi = \frac{1}{x}\) directs focus to infinity, the point is labeled a regular singular point if:

• At least one of the functions from the ordinary point's condition does not have a Taylor series expansion about \(\xi = 0\).
• Both of the following functions do have Taylor series expansions:
  • \( \frac{\xi}{P(1 / \xi)}\left[\frac{2 P(1 / \xi)}{\xi} - \frac{Q(1 / \xi)}{\xi^{2}}\right] \)
  • \( \frac{R(1 / \xi)}{\xi^{2} P(1 / \xi)} \)

This indicates a more intricate, but still feasible, solution approach is needed, often leveraging Frobenius or other specialized methods, potentially involving series with non-integer powers or logarithmic terms.

Taylor Series Expansion

Taylor series expansion is a powerful tool in calculus to approximate functions that are infinitely differentiable at a given point. The idea is to represent a function as an infinite sum of its derivatives evaluated at a point:\[ f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots \]For a differential equation to have a point as ordinary or regular singular, the ability to express parts of the equation as Taylor series around a point (like \(\xi = 0\)) is crucial. This ensures:

• Predictable behavior of functions involved.
• Simplified solution methods using series solutions.

Taylor series make it possible to explore and solve complex differential equations by converting them into series forms, thereby enabling easier manipulations and integrations, even around potential points of singularity. Not only does this make solving equations feasible, but it also provides deeper insights into function behavior across different regions.