Question

In each exercise, find the singular points (if any) and classify them as regular or irregular. $$ \left(t^{2}-1\right)^{2} y^{\prime \prime}+(t+1) y^{\prime}+y=0 $$

Short answer

Question: Determine the singular points of the given second-order linear differential equation and classify them as regular or irregular: $$ \left(t^{2}-1\right)^{2} y^{\prime \prime}+(t+1) y^{\prime}+y=0 $$ Answer: The singular points are at \(t=1\) and \(t=-1\), and both are regular singular points.

Step-by-step solution

Rewrite the equation in standard form

The differential equation is given by: $$ \left(t^{2}-1\right)^{2} y^{\prime \prime}+(t+1) y^{\prime}+y=0 $$ We can rewrite it in the standard form by dividing the equation by \(\left(t^2 - 1\right)^2\): $$ y^{\prime \prime}+\frac{t+1}{\left(t^2-1\right)^{2}} y^{\prime}+\frac{1}{\left(t^2-1\right)^{2}}y=0 $$

Identify the singular points

Singular points occur at the values of \(t\) that make the coefficients of \(y'\) and \(y\) have a singularity. In this case, we analyze the denominators of the coefficients: 1. For the coefficient of \(y'\): $$ \left(t^2-1\right)^{2} = (t-1)^2 (t+1)^2 $$ The singular points occur at \(t=1\) and \(t=-1\). 2. For the coefficient of \(y\): $$ \left(t^2-1\right)^{2} = (t-1)^2 (t+1)^2 $$ The singular points also occur at \(t=1\) and \(t=-1\). So, the singular points of the given differential equation are \(t=1\) and \(t=-1\).

Classify the singular points as regular or irregular

A singular point is regular if both the coefficient of \(y'\) and the coefficient of \(y\) have finite values when expanded in power series, and it's irregular otherwise. Let's analyze the coefficients at the singular points: 1. At \(t=1\): $$ \frac{t+1}{\left(t^2-1\right)^{2}} = \frac{2}{(2)^{2}} = \frac{1}{2} $$ $$ \frac{1}{\left(t^2-1\right)^{2}} = \frac{1}{(2)^{2}} = \frac{1}{4} $$ Both coefficients have finite values at \(t=1\), so the singular point \(t=1\) is a regular singular point. 2. At \(t=-1\): $$ \frac{t+1}{\left(t^2-1\right)^{2}} = \frac{0}{(2)^{2}} = 0 $$ $$ \frac{1}{\left(t^2-1\right)^{2}} = \frac{1}{(2)^{2}} = \frac{1}{4} $$ Both coefficients have finite values at \(t=-1\), so the singular point \(t=-1\) is a regular singular point. Therefore, both singular points \(t=1\) and \(t=-1\) are regular singular points.

Key concepts

Regular Singular Point

A regular singular point is an essential concept when analyzing differential equations. It's a type of singularity that dictates the strategy used to find a solution to the equation. In a differential equation, a point is considered regular singular if the solution around that point can be expressed as a power series with possibly a finite number of terms of negative powers. To determine this, we often look at the coefficients in the equation when written in a standard form.

In the step-by-step solution provided, we see that the points where the coefficients of the derivatives become infinite are checked. After a reformulation of the coefficients at these points, we concluded that the points \(t=1\) and \(t=-1\) are regular singular points because the coefficients of \(y'\) and \(y\) expanded in power series around these points are finite. This characteristic is vital because it enables us to use specific methods, like the Frobenius method, to determine the solution in a series form close to these points.

Understanding regular singular points is fundamental for students to grasp how to approach the given differential equation and what methods will lead to a valid solution.

Irregular Singular Point

On the flip side, an irregular singular point is where the behavior of the differential equation's solutions is less 'tame' as compared to regular singular points. If expanded around these points, we find that the coefficients of the derivatives of the solution do not result in a power series representation with a finite number of negative powers. When approaching an irregular singular point, the solutions might involve more complex functions and series, such as exponential or asymptotic series.

For the exercise in question, the solution identified no irregular singular points, as the points \(t=1\) and \(t=-1\) were found to be regular. However, if they had been classified as irregular, it would imply that the solutions near these points would not be analyzable by the usual Frobenius or power series methods. This distinction highlights the divergent nature of solutions near different types of singularities, and it reinforces the need for alternative analytical or numerical methods to tackle equations with irregular singularities.

Second-Order Differential Equation

A second-order differential equation is an equation that involves the unknown function \(y\), its first derivative \(y'\), and its second derivative \(y''\). The degree of the highest derivative in the equation classifies the order of a differential equation. In the study of differential equations, second-order ones are particularly important due to their appearance in numerous physical applications, including mechanics, oscillations, and electrical circuits.

The exercise we're examining features a second-order differential equation with singular points. Students often encounter these equations in the context of initial value or boundary value problems, where additional information is provided to solve the equation uniquely. By converting to standard form and identifying the nature of its singular points, one can decide on the appropriate techniques to solve these equations, like the power series or Frobenius methods for equations with regular singular points.

Power Series Method

Finally, the power series method is a powerful analytical tool used to tackle differential equations, especially around singular points. This method assumes a solution can be represented as a power series expansion. When applied to a differential equation, the power series method involves substituting a series expression for \(y\) into the equation and matching coefficients for each power of \(t\) to determine the series coefficients.

In the context of our exercise, the power series method could be employed to find solutions around the regular singular points, \(t=1\) and \(t=-1\). Through this method, we convert the daunting task of solving a differential equation into a more tractable one – finding the coefficients of a power series. This transformation to an algebraic problem not only simplifies the process but also allows for a clear understanding of the behavior of solutions near specific points.

Using the power series method offers an insightful way to approach and solve second-order differential equations with regular singular points, and it's a staple in the toolkit for students studying advanced mathematics or physics.