Question

Find all singular points of the given equation and determine whether each one is regular or irregular. \(x^{2}(1-x)^{2} y^{\prime \prime}+2 x y^{\prime}+4 y=0\)

Short answer

The singular points are x = 0 and x = 1. 2. Are the singular points regular or irregular? Both singular points, x = 0 and x = 1, are regular singular points.

Step-by-step solution

Rewrite the equation in standard form

To rewrite the given differential equation in its standard form, we first need to divide the entire equation by the leading coefficient \(x^2(1-x)^2\). The given equation is: $$x^{2}(1-x)^{2} y^{\prime \prime}+2 x y^{\prime}+4 y=0$$ Dividing by \(x^2(1-x)^2\), we obtain the standard form: $$y^{\prime \prime} + \frac{2x}{x^2(1-x)^2}y^{\prime} + \frac{4}{x^2(1-x)^2}y = 0$$

Identify the singular points

A singular point is a value of the independent variable \(x\) where the coefficients of \(y^\prime\) and \(y\) become infinite or undefined. To find the singular points, let's examine the coefficients of \(y^\prime\) and \(y\) in the standard form of the equation: Coefficient of \(y^\prime\): \(\frac{2x}{x^2(1-x)^2}\) Coefficient of \(y\): \(\frac{4}{x^2(1-x)^2}\) The coefficients become undefined when the denominators are equal to zero, i.e., when \(x=0\) or \(1-x = 0\), which leads to \(x=1\). Therefore, we have two singular points: \(x = 0\) and \(x = 1\).

Determine if the singular points are regular or irregular

To determine if a singular point is regular or irregular, we can check the conditions for a regular singular point. For a singular point to be regular, both of the following quantities are required to be analytic (have a convergent power series expansion) at the singular point: 1. \((x-x_0)P(x)\), and 2. \((x-x_0)^2 Q(x)\) where \(x_0\) is the singular point and \(P(x)\) and \(Q(x)\) are the coefficients in the standard form of the differential equation: $$y^{\prime \prime} + P(x)y^{\prime} + Q(x)y = 0$$ We have \(P(x) = \frac{2x}{x^2(1-x)^2}\) and \(Q(x) = \frac{4}{x^2(1-x)^2}\). For the singular point \(x_0 = 0\): 1. \((x-0)P(x) = x \cdot \frac{2x}{x^2(1-x)^2} = \frac{2x}{(1-x)^2}\) which is analytic at \(x=0\). 2. \((x-0)^2 Q(x) = x^2 \cdot \frac{4}{x^2(1-x)^2} = \frac{4}{(1-x)^2}\) which is analytic at \(x=0\). For the singular point \(x_0 = 1\): 1. \((x-1)P(x) = (1-x) \cdot \frac{2x}{x^2(1-x)^2} = \frac{2x}{x^2(1-x)}\) which is analytic at \(x=1\). 2. \((x-1)^2 Q(x) = (1-x)^2 \cdot \frac{4}{x^2(1-x)^2} = \frac{4}{x^2}\) which is analytic at \(x=1\). Therefore, both singular points \(x = 0\) and \(x = 1\) are regular singular points.

Key concepts

Regular Singular Point

When analyzing differential equations, certain types of singular points are characterized by how the functions within the equation behave near these points. A regular singular point is quite special. It is defined as a point where the solution to the differential equation might be infinite or not well-defined, but still has a certain predictable structure.

In the context of the second-order linear differential equation \[y'' + P(x)y' + Q(x)y = 0\], a point \(x_0\) is considered a regular singular point if the functions \((x-x_0)P(x)\) and \((x-x_0)^2Q(x)\) are analytic at \(x_0\). Being analytic means that the functions can be expanded into convergent power series around the point. This property is immensely valuable as it allows us to use series solutions to find the behavior of the solution near this point.

For example, in the equation given by the textbook, the singular points at \(x=0\) and \(x=1\) meet these criteria, indicating that they are regular singular points. It is important to note that the presence of regular singular points is a cue that the solution may involve the Frobenius method, a technique for finding power series solutions to differential equations.

Irregular Singular Point

In contrast to regular singular points, an irregular singular point is one that does not constrain the behavior of solutions nearly as much. In our standard form differential equation \(y'' + P(x)y' + Q(x)y = 0\), if \((x-x_0)P(x)\) and \((x-x_0)^2Q(x)\) fail to be analytic at some point \(x_0\), that point is classified as an irregular singular point.

While regular singular points allow for solutions to be expressed in terms of power series, irregular singular points do not offer this neat structure. Solutions at these points may exhibit behavior that is more complicated, potentially involving terms growing faster than any power of \(x\).

In our given problem, fortunately, we only encountered regular singular points. If we had discovered that either of the functions \((x-x_0)P(x)\) or \((x-x_0)^2Q(x)\) were not analytic, then we would have had to declare the corresponding \(x_0\) an irregular singular point, which would demand a different analytical approach to solve.

Standard Form Differential Equation

Understanding the standard form of a differential equation is crucial because it simplifies the process of analysis and solution. The standard form of a second-order linear differential equation is given by \(y'' + P(x)y' + Q(x)y = 0\), where \(P(x)\) and \(Q(x)\) are functions of \(x\). By converting a differential equation into this form, singularities are made explicit, and regularity conditions can be easily applied.

To rewrite an equation in standard form, one typically divides each term by the highest order coefficient that depends on the independent variable, which in our exercise was \(x^2(1-x)^2\). This manipulation not only gives us a clearer view of the behavior of the equation near potential singularities but also sets up the groundwork for applying further solution techniques, such as looking for power series solutions around singular points.

Analytic Function

An analytic function is a function that can be represented by a convergent power series around a certain point within its domain. In simpler terms, if you can write the function as an infinite sum of powers of \((x-a)\) for some point \(a\), and this series converges to the function’s values when \(x\) is near \(a\), then the function is analytic at \(a\).

Analyticity is a central concept when examining the behavior of differential equations at singular points. Analytic functions have the nice property of being differentiable as many times as we like, which means their derivatives also have power series expansions. This makes them particularly conducive to the Frobenius method for finding series solutions around regular singular points.

Any polynomial, exponential function, sine, and cosine are examples of analytic functions. When we looked at the functions \((x-x_0)P(x)\) and \((x-x_0)^2Q(x)\) in the exercise, proving they were analytic at the singular points was a key step in identifying the nature of these points.