Question

Find all singular points of the given equation and determine whether each one is regular or irregular. \((\sin x) y^{\prime \prime}+x y^{\prime}+4 y=0\)

Short answer

Question: Identify all the singular points of the given differential equation \((\sin x) y^{\prime \prime}+x y^{\prime}+4 y=0\) and determine if each singular point is regular or irregular. Answer: The differential equation has singular points at \(x=k\pi\), where \(k\in\mathbb{Z}\). All these singular points are regular.

Step-by-step solution

Identify Singular Points

To find the singular points, we need to examine the coefficients of the differential equation. Observe that the coefficients are $$ p(x) = \frac{x}{\sin x} \quad\text{and}\quad q(x) = \frac{4}{\sin x}. $$ A point \(x_0\) will be a singular point if any of the coefficients \(p(x)\) or \(q(x)\) is not analytic in its neighborhood. In order to do this, we will look for any values of x for which either \(p(x)\) or \(q(x)\) is undefined. We have \(p(x)\) and \(q(x)\) undefined when \(\sin x=0\). This occurs at \(x=k\pi\), where \(k\in\mathbb{Z}\).

Determine if Singular Points are Regular or Irregular

Now that we have identified the singular points, we need to determine if each singular point is regular or irregular. A singular point \(x_0\) is regular if the functions \((x-x_0)p(x)\) and \((x-x_0)^2q(x)\) are analytic in its neighborhood. Let's consider a general singular point \(x_0=k\pi\), where \(k\in\mathbb{Z}\). Then, we examine the functions \((x-x_0)p(x)\) and \((x-x_0)^2q(x)\): \((x-x_0)p(x) = (x-k\pi)\frac{x}{\sin x} \quad\text{and}\quad (x-x_0)^2q(x) = (x-k\pi)^2\frac{4}{\sin x}\). Both functions are analytic in the neighborhood of any \(k\pi\), except at the point \(k\pi\). In other words, these functions have removable singularities at these points. We can use the L'Hopital's rule to determine the limits of these functions at \(k\pi\). - For \((x-x_0)p(x)\), we have the limit: $$ \lim_{x\rightarrow k\pi}(x-k\pi)\frac{x}{\sin x}, $$ which can be evaluated using L'Hopital's rule, giving us \(\lim_{x\rightarrow k\pi} \frac{x}{\cos x}\). Since \(\cos x\) is never zero in the neighborhood of \(x=k\pi\), this limit exists and the function is analytic at \(x=k\pi\). - Similarly, for \((x-x_0)^2q(x)\), we have the limit: $$ \lim_{x\rightarrow k\pi}(x-k\pi)^2\frac{4}{\sin x}. $$ Applying L'Hopital's rule two times, we obtain \(\lim_{x\rightarrow k\pi}-x^3\sin x\). This limit also exists and the function is analytic at \(x=k\pi\). Since both \((x-x_0)p(x)\) and \((x-x_0)^2q(x)\) are analytic at all singular points \(x=k\pi\), we can conclude that all these singular points are regular.

Summary

We found that the given differential equation has singular points at \(x=k\pi\), where \(k\in\mathbb{Z}\). We then determined that each singular point is regular as the functions \((x-x_0)p(x)\) and \((x-x_0)^2q(x)\) are analytic at all these singular points, by applying L'Hopital's rule.

Key concepts

Regular Singular Point

When solving differential equations, identifying singular points provides insights into the nature of the solutions near those points. A singular point of a differential equation is a value of the independent variable where the coefficients of the highest derivatives become infinite or undefined. Among singular points, regular ones have a special significance.

A regular singular point, denoted as the value \( x_0 \), is one where the function \( p(x) \) - the coefficient of \( y' \) - and the function \( q(x) \) - the coefficient of \( y \) - become undefined when the highest derivative's coefficient vanishes. However, for a point to be classified as regular, the functions \( (x-x_0)p(x) \) and \( (x-x_0)^2q(x) \) must be analytic around \( x_0 \). Analytic functions, which we will discuss shortly, are those that are well-behaved and have series expansions.

In the exercise, after identifying singular points at \( x=k\text{π} \) by finding where the sine function vanishes, the step-by-step solution showed that the functions multiplied by \( (x-x_0) \) and \( (x-x_0)^2 \) remained analytic, except at the singular points. Using L'Hopital's rule, the removeable singularities were resolved, confirming each singular point as regular. These points are crucial since they influence the form of potential solutions and dictate the methods used for finding them. Understanding this delineation helps students to select appropriate techniques for solving complex differential equations.

Analytic Function

In the realm of differential equations, being familiar with analytic functions is essential. An analytic function is one that can be expressed as a power series — a sum of terms with increasing powers — which converges to the function's values within some radius of convergence around a point \( x_0 \). Essentially, an analytic function behaves well under differentiation and integration, and its local behavior can be captured by an infinite polynomial.

What distinguishes analytic functions in the discussion of differential equations, particularly at singular points, is their predictability. Since an analytic function has derivatives of all orders, it lends itself to precise approximations using its Taylor series near the point \( x_0 \). This feature allows us to study the behavior of differential equation solutions around singular points.

The exercise involved checking the analyticity of certain functions to determine the nature of singular points. Doing so, we affirm that near each regular singular point, the solutions of the differential equation can be represented by convergent series expansions. This understanding of analytic functions is a cornerstone in the theory of ordinary differential equations, enabling students to gain insights into the nature of the solutions and the behavior of the equation under various conditions.

L'Hopital's Rule

In certain mathematical problems, determining the limit of a function as it approaches a particular point can be challenging, especially when facing indeterminate forms like 0/0 or \( \infty/\infty \). L'Hopital's rule is the key to unlocking these limits. It asserts that if the limits of two functions \( f(x) \) and \( g(x) \) approach \( 0 \) or \( \infty \) as \( x \) approaches \( a \), and their derivatives are known at that point, then the original limit can be found by calculating the limit of the ratio of their derivatives instead.

In the exercise, L'Hopital's rule was applied to the functions arising after multiplying \( p(x) \) and \( q(x) \) by the terms \( (x-x_0) \) and \( (x-x_0)^2 \). By doing so, the solution demonstrated the analyticity at regular singular points, which initially presented indeterminate forms. Students learn through this rule that complex limits are not impasses but puzzles that can be solved, fostering deeper understanding of limits and continuous functions. This technique is not only invaluable for differential equations but also for a broad range of problems across calculus and higher mathematics.