Question

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

Short answer

Answer: The singular points of the given equation are \(x = 0, \pm 1\). The singular points \(x= \pm 1\) are regular, while the singular point \(x=0\) is irregular.

Step-by-step solution

Identify Singular Points

First, we need to find the singular points of the given equation by identifying the points where the coefficients of \(y''\) and \(y'\) become undefined or infinite. For the coefficient of \(y''\), we have the term \(x^2(1-x^2)\). This term becomes undefined when \(x^2=0\) or \(x^2=1\), meaning the singular points are \(x=0\) and \(x=\pm1\). For the coefficient of \(y'\), we have the term \((2/x)\). This term becomes undefined when \(x=0\), which is already a singular point from our previous analysis. So, the singular points of the given equation are \(x = 0, \pm 1\).

Determine Regular or Irregular Singular Points

Now, we need to check whether these singular points are regular or irregular. For a singular point to be regular, the coefficients of lower-order derivatives must have converging power series around the singular point. If not, the singular point is irregular. For \(x=\pm1\), we can write the coefficients of the derivatives in the following form: - Coefficient of \(y''\): \((1-x^2) = (1-1)^2 = 0\) and \((1-(-1)^2) = 0\). These coefficients vanish at the singular points and therefore also converge. - Coefficient of \(y'\): \((2/x) = 2\) for \(x = 1\), and \(-2\) for \(x = -1\). Both of these are finite and converging around their respective singular points. Therefore, the singular points \(x= \pm 1\) are regular singular points. Now, let's examine the singular point at \(x=0\): - Coefficient of \(y''\): \((1-x^2) = 1\). This coefficient does not vanish at the singular point but is finite and converging. - Coefficient of \(y'\): \((2/x)\). This term becomes infinite as \(x\) approaches zero. We can find its limit: \(\lim_{x\to 0}\frac{2}{x}=\infty\) Since the coefficient of \(y'\) becomes infinite at \(x=0\), the singular point at \(x=0\) is irregular. So, the given equation has one irregular singular point, \(x=0\), and regular singular points at \(x=\pm 1\).

Key concepts

Regular Singular Point

A regular singular point is a special type of singular point in a differential equation where the behavior of the solution near the singularity is well-behaved in a certain sense. This means that, even though the coefficients may become undefined at the singularity, they do so in a predictable way. A singular point, say at \( x_0 \), is considered regular if:

• The coefficients of the lower-order terms have a power series expansion around \( x_0 \) that converges.
• The solution to the differential equation near \( x_0 \) can be expressed in a series form that converges.

In the given problem, the singular points \( x = \pm 1 \) are regular singular points. This is because the coefficients, though vanishing at these points, allow the perturbations to converge in a neat power series. This makes it possible to find solutions that behave well around \( x = \pm 1 \). Understanding regular singular points helps in using methods like the Frobenius method to find exact solutions.

Irregular Singular Point

Unlike regular singular points, irregular singular points are where the coefficients of a differential equation become undefined or non-convergent in a more chaotic manner. At an irregular singular point, the behavior of solutions can be unpredictable or complex. For a singularity at \( x_0 \) to be irregular:

• The coefficients of the equation do not have a convergent power series expansion about \( x_0 \).
• The solutions may not be representable in a convergent series form.

In our example, the point \( x = 0 \) is identified as an irregular singular point because as \( x \) approaches zero, the coefficient \( \frac{2}{x} \) becomes infinite. This infinite behavior means that the solutions can exhibit erratic behavior that is difficult to model uniformly. Understanding whether a point is irregular is crucial in determining the appropriate methods to solve or approximate solutions of the differential equation.

Differential Equation

Differential equations are mathematical equations that relate a function with its derivatives. They describe how a particular quantity changes over time or space and are fundamental in modeling various real-world phenomena, such as physics, engineering, biology, and economics. A second-order differential equation, like the one in our problem, typically involves the function \( y \), its first derivative \( y' \), and its second derivative \( y'' \).In this exercise, the given differential equation is:\[ x^2(1-x^2) y'' + \frac{2}{x} y' + 4y = 0 \]Here, the terms involve polynomial expressions in \( x \) and inverse powers of \( x \). Singular points are specific values of \( x \) where these terms become undefined or problematic. Analyzing these points helps us understand how solutions behave and ensures that essential conditions like continuity and differentiability are maintained. Mastery of differential equations aids in solving complex problems across various scientific disciples.

Coefficient Convergence

Coefficient convergence is an essential concept when determining the nature of singular points in differential equations. It refers to whether the coefficients of the differential equation expand into a power series that converges. For singular points:

• If coefficients converge to a power series in the neighborhood of the singular point, the point can be regular.
• If they diverge or do not form a series, it can be irregular.

When dealing with regular singular points, it's key that the series expansions of coefficients are finite and well-defined, allowing for predictable solution behavior. For our problem, examining how terms like \( (1-x^2) \) and \( \frac{2}{x} \) behave at different \( x \) values tells us about the type of singularity. Coefficient convergence becomes a core part of techniques like series solutions, where constructing a suitable series form requires these coefficients to behave in a convergent manner that supports the desired continuity and differentiability of the solutions.