Question

Find all the regular singular points of the given differential equation. Determine the indicial equation and the exponents at the singularity for each regular singular point. \(y^{\prime \prime}+4 x y^{\prime}+6 y=0\)

Short answer

Answer: No, there are no regular singular points in the given differential equation.

Step-by-step solution

Identify the regular singular points

To find the singular points, we must rewrite the differential equation in the standard form, i.e. when all the coefficients are rational functions. Divide through by the leading coefficient to get the equation to the form: \((1)y^{\prime \prime} + \frac{4 x}{1}y^{\prime} + \frac{6}{1}y = 0\). Now the equation has the form \(y^{\prime \prime }+P(x) y^{\prime}+Q(x) y = 0\), where \(P(x) = \frac{4x}{1}\) and \(Q(x) = \frac{6}{1}\). The regular singular points are the points where the functions \(P(x)\) and \(Q(x)\) have poles. Since both \(P(x)\) and \(Q(x)\) are rational functions (in fact, they are both polynomials), we can simply look for points where the denominators vanish. In this case, there are no such points; thus, there are no regular singular points. In this particular case, there are no regular singular points to analyse, so the problem is solved.

Key concepts

Differential Equations

Differential equations are mathematical equations that involve derivatives of a function. They are used to represent the rate of change of one variable with respect to another. In essence, a differential equation will give you a relation between a function and its derivatives.

Let's break this down. Every differential equation has an unknown function, typically denoted by letters such as 'y' or 'u', and its derivatives, like 'y^{\textprime}' for the first derivative or 'y^{\textprime\textprime}' for the second derivative. The exercise at hand, \(y^{\textprime \textprime}+4 x y^{\textprime}+6 y=0\), is a second-order linear homogeneous differential equation. This means the equation relates the function \(y\), its first derivative \(y^{\textprime}\), and its second derivative \(y^{\textprime\textprime}\) without any terms that are not multiples of \(y\) or its derivatives. Homogeneous indicates that there's no separate function of \(x\) by itself, i.e., terms without \(y\).

Understanding differential equations is crucial as they are used to model real-world phenomena in physics, engineering, economics, and many other fields. The solutions to these equations help predict the behavior of dynamic systems over time.

Indicial Equation

When analyzing differential equations, especially those with singular points, one might encounter the term 'indicial equation'. This special type of equation arises when solving linear differential equations around a regular singular point, which is discussed in the 'singular point analysis' section.

An indicial equation provides you with possible values of an index (or exponent) that relate to the behavior of the solution of the differential equation near a singular point. In more mathematical terms, if you're looking for a solution of the form \(x^r\) near a singular point, where \(r\) is unknown, you'll use the indicial equation to solve for it.

As the provided exercise illustrates, \(y^{\textprime \textprime}+4 x y^{\textprime}+6 y=0\) has no regular singular points. Therefore, the step of formulating an indicial equation doesn't apply here. However, in cases where there are regular singular points, finding the roots of the indicial equation is a critical step towards the full solution of the differential equation around those points.

Singular Point Analysis

Singular point analysis deals with those points in a differential equation where the usual analytic solution methods might fail - specifically at points where the coefficients of the derivatives exhibit singular behavior, such as being undefined or infinite.

In the provided exercise, the task was to identify any 'regular singular points.' Regular singular points are particular types of singularities around which a differential equation can still be solved using power series methods, although special techniques, like the Frobenius method, may be required. These points are characterized by having coefficients in the equation that are ratios of polynomials (rational functions) where the pole (a point where the function goes to infinity) is of a certain order.

To determine if a differential equation has regular singular points, we look at the coefficients of the derivatives \(P(x)\) and \(Q(x)\) and identify points where they become infinite. For the given differential equation, \(P(x) = 4x\) and \(Q(x) = 6\) both present as polynomials, which means they are well-behaved everywhere (no poles), and thus no regular singular points occur.

While this concept may sound abstract, it is vital for understanding the nature of solutions to differential equations, especially those that may initially appear problematic or unsolvable. In physical systems, these solutions often describe the behavior near critical points – such as near the center of a star or at the tip of a crack in materials.