Question

Find all singular points of the given equation and determine whether each one is regular or irregular. \((x \sin x) y^{\prime \prime}+3 y^{\prime}+x y=0\)

Short answer

The singular points of the given differential equation are x=0 and x=kπ (where k is an integer). Both singular points are regular.

Step-by-step solution

Find the singular points

To find the singular points, we set the coefficient of \(y''\) to zero. In our equation, the coefficient of \(y''\) is \((x \sin x)\). Set it to zero and solve for x: \(x \sin x = 0\) There are two possible singular points: 1. \(x=0\), since \(0 \times \sin(0) = 0\) 2. \(x=k\pi\), where \(k\) is an integer, since \(\sin(x)\) is 0 at its multiples of \(\pi\) Now, we will analyze the behavior of the differential equation around these singular points.

Analyze the behavior of the differential equation around x=0

To determine whether the singular point at x=0 is regular or irregular, we should investigate how the coefficients of the differential equation behave around this point. Consider the coefficients of \(y''\), \(y'\), and \(y\): 1. For \(y''\), we have \((x \sin x)\), which is equal to \(0\) at x=0 2. For \(y'\), we have the constant coefficient 3, which does not behave badly at x=0 3. For \(y\), we have \(x\), which is well-behaved around x=0 Since all the coefficients are well-behaved around x=0, the singular point is regular.

Analyze the behavior of the differential equation around x=k\(\pi\)

We will now determine if the singular point x=k\(\pi\) is regular or irregular. As in step 2, we will analyze the behavior of the coefficients around x=k\(\pi\): 1. For \(y''\), we have \((x \sin x)\), which is equal to \(0\) at x=k\(\pi\) 2. For \(y'\), we have the constant coefficient 3, which does not behave badly at x=k\(\pi\) 3. For \(y\), we have \(x\), which is well-behaved around x=k\(\pi\) Since all the coefficients are well-behaved around x=k\(\pi\), the singular point is regular.

Conclusion

All the singular points of the given differential equation, x=0 and x=k\(\pi\), are regular.

Key concepts

Understanding Regular and Irregular Singular Points

In differential equations, a singular point is a value of the independent variable where the coefficients cause the equation to lose its normal meaning. Such points serve as gateways to deepening our awareness of a given equation's intricacies.

Regular singular points are locations where, despite the coefficient function's unusual behavior (usually becoming zero), the equation retains a predictable structure. Here, the solution to the equation near the singular point can generally be expressed in a power series.

Irregular singular points, on the other hand, lead to solutions that defy power series description. These solutions might include exponential or other non-polynomial behavior, signaling a more complex situation that typically demands advanced analytical methods.

In our exercise, singular points were identified at \(x=0\) and \(x=k\textstyle\frac{\textcolor{blue}{\text{\textpi}}}{\textcolor{white}{\text{\textpi}}}\textcolor{black}{\text{\textpi}}\), where \(k\) is an integer. By analyzing the behavior of the coefficients, we concluded that these points are regular singular points because the equation remains well-behaved around these values.

Navigating Through Differential Equation Coefficients

The coefficients in a differential equation play a pivotal role in understanding the behavior of solutions. These coefficients, often functions of the independent variable(s), can suggest the stability, periodicity, and other features of solutions.

In simpler terms, if we think of a differential equation as a recipe, the coefficients are like the critical spices that define the dish's final flavor. They help to determine the growth, decay, or oscillations in the equation's solutions.

When solving a differential equation, spotting where these coefficients become zero or infinity is crucial, as they point us to the singular points. The exercise we examine pivots on the behavior of these coefficients around the identified singular points, with regular behavior indicating regular singular points.

To determine the nature of these points, as in the exercise, one typically investigates whether the coefficient functions and their derivatives, when standardized, remain bounded near the singular points. If they do, the points are deemed regular; otherwise, irregular.

Unraveling Boundary Value Problems

Moving away from the singularities of differential equations, we encounter boundary value problems (BVPs), which play a crucial role in engineering, physics, and other applied sciences. These problems aim to find a solution to a differential equation that also satisfies certain specified conditions at the boundaries of the domain.

In a boundary value problem, you are given not just the equation to solve, but also information about the solution's behavior at the 'edges' of the considered interval. For example, we might know the solution's value at the beginning and end of the time period or space interval.

BVPs contrast with initial value problems (IVP), where one typically knows the 'starting conditions.' While IVPS are like telling a story with a clear beginning, BVPs are akin to puzzles where both the beginning and the end need to fit the picture just right. In the context of our exercise, while a boundary value problem was not explicitly solved, understanding BVPs is valuable for deciphering many real-world scenarios where differential equations are applicable.