Question

In each exercise, find the singular points (if any) and classify them as regular or irregular. $$ t^{2} y^{\prime \prime}+(\sin t) y^{\prime}+y=0 $$

Short answer

Question: Identify the singular points of the second-order linear differential equation \(t^2y'' + (\sin t)y'+y=0\) and classify them as regular or irregular. Answer: The given differential equation has a singular point at \(t=0\), which is a regular singular point.

Step-by-step solution

Identify the coefficients of the differential equation

In the given equation: \(t^2y'' + (\sin t)y'+y=0\), the coefficients are as follows: - \(p(t) = \frac{\sin t}{t^2}\) - \(q(t) = \frac{1}{t^2}\)

Identify singular points

To determine if there are any singular points, we need to check if the coefficients are analytic (having a convergent power series representation) for all values in the given interval. In this case, we see that both \(p(t)\) and \(q(t)\) are not analytic when \(t=0\). Therefore, there is a singular point at \(t=0\).

Classify the singular point

Next, we need to classify the singular point as regular or irregular. If \(t=0\) is a regular singular point, then we should be able to find a positive integer \(N\) such that \((t-0)^Np(t)\) and \((t-0)^{N+1}q(t)\) are analytic at the singular point. Here's the test for regularity: - \((t-0)^Np(t) = t^N\frac{\sin t}{t^2}\) - \((t-0)^{N+1}q(t) = t^{N+1}\frac{1}{t^2}\) Now, we need to determine if there exists a positive integer N, such that both of these functions are analytic in a neighborhood of \(t=0\). Let's choose N=2: - \((t-0)^2p(t) = t^2\frac{\sin t}{t^2} = \sin t\) - \((t-0)^3q(t) = t^3\frac{1}{t^2} = t\) Since both \(\sin t\) and \(t\) are analytic in a neighborhood of \(t=0\), we can conclude that \(t=0\) is a regular singular point.

Key concepts

Regular Singular Points

When examining differential equations like the one in the exercise,understanding singular points is crucial. A regular singular point is a location where the functions involved in the equation stop being analytic, but with some "wobbly" behavior that is predictable and manageable.

In a differential equation, such as the one given,we check the coefficients of the terms to identify potential singular points. We specifically look for places where these coefficients may not be analytic.

For a point to be regular,there must exist a positive integer \(N\) such that multiplying the coefficient term by a power of the singular point makes it analytic again. In our step-by-step solution,the coefficients from the exercise had to be handled in such a manner.In the end,by finding \(N=2\),the singularity at \(t=0\) was reconciled as a regular singular point.

A regular singular point allows solutions to be found in terms of a power series,which is immensely useful in understanding the behavior of systems at those points.

Irregular Singular Points

Irregular singular points are a bit trickier than regular ones. While regular singular points can be managed and understood through power series, irregular singular points tend to defy neat solutions. They occur when such adjustments as in regular singularities (i.e., multiplying by powers as in the previous section) still don't salvage the analyticity of the coefficients.

This means no simple power series around those points exist, which presents challenges in formally solving the equation near those spots. Irregular points lead to solutions that can involve series that are divergent or complex exponential in nature. This makes analysis at these points more complex and often requires advanced analytical techniques.

Analytic Functions

Finding analyticity in a function around a point means that we can express the function as a convergent power series there.This is an essential concept when determining the nature of singular points in differential equations,as shown in the exercise.For a function to be analytic,it continuously differentiates all the way around a given point,and we can write it as a sum of power terms centred on that point.

To check if a function is analytic,consider performing a power series expansion near the point in question.For example,functions like \(\sin t\),which appear in the problem's coefficients,are analytic everywhere,including around \(t=0\).This property was crucial in determining if the singular point in our exercise was regular.

Power Series Representation

The power series representation is a cornerstone in solving differential equations, especially around singular points.A power series is an infinite sum of the form:\[ f(t) = \sum_{n=0}^{\infty} a_n (t - t_0)^n \]where \(a_n\) are coefficients, and \(t_0\) is the point around which the series is expanded.

In our exercise,this representation helps in determining if the singularity is regular by expressing coefficients in such a series.When \(p(t)\) and \(q(t)\) can be converted into such series around a point,solutions in the form of power series are often obtainable. This is not only a tool for simplification but also ensures the solutions behave predictably. Regular singular points, when transformed appropriately, can yield clean and manageable series solutions, which are vital for further mathematical analyses or applications.