Question

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

Short answer

Answer: The given equation has one singular point, \(x=0\), and this point is irregular.

Step-by-step solution

Find singular points

To find all singular points, we analyze the coefficients of the highest and lowest derivatives and check which of them become infinite or discontinuous. We have the following coefficients: Coefficient of the highest derivative (second-order derivative): 1 Coefficient of the lowest derivative (first-order derivative): \(\ln|x|\) Notice that the coefficient of the second derivative is constant (1) and doesn't have any infinite or discontinuous values. Now, we check for the singular points in the coefficient of the first derivative: \(\ln|x|\) The natural logarithm is discontinuous and undefined at \(x=0\). Therefore, \(x=0\) is a singular point.

Determine if the singular point is regular or irregular

We need to analyze the singular point \(x=0\) to determine whether it is regular or irregular. For this, we'll verify if it satisfies the following conditions: 1. The coefficient \(\frac{(\ln|x|)}{x}\) has a limit at \(x=0\). 2. The coefficient \(\frac{3x}{x^2}\) has a limit at \(x=0\). Let's find the limit of the coefficients at \(x=0\): 1. \(\lim_{x\to0}\frac{(\ln|x|)}{x}\): As x approaches 0, the natural logarithm approaches \(-\infty\), and as a result, the fraction becomes indeterminate. We can use L'Hopital's Rule to further analyze this limit: \(\lim_{x\to0}\frac{(\ln|x|)}{x} = \lim_{x\to0}\frac{(\frac{1}{x})}{1} = \lim_{x\to0}(\frac{1}{x})\) This limit clearly goes to \(\infty\) as x approaches 0, so the first condition is not satisfied. Since the first condition didn't hold, we don't need to check the second condition. Therefore, the singular point \(x=0\) is irregular. Hence, the given equation has only one singular point, \(x=0\), and this point is irregular.

Key concepts

Regular Singular Point

When studying differential equations, identifying the nature of singular points is essential for understanding the behavior of solutions near these points. A regular singular point occurs in a second-order linear differential equation of the form
\[\begin{equation}y^{\prime \prime} + p(x)y^{\prime} + q(x)y = 0 \end{equation}\]
where the functions p(x) and q(x) have singularities.

To determine if a singular point, say x = x_0, is regular, we investigate the limits of p(x)(x - x_0) and q(x)(x - x_0)^2 as x approaches x_0. If both limits exist, then x_0 is a regular singular point. This concept is crucial because it dictates which method we can use to find a solution around x_0. Regular singular points allow for solutions to be expressed in the form of a power series, which might include a logarithmic term only if the point is indeed regular singular.

Irregular Singular Point

Conversely, an irregular singular point refers to a singularity where the mathematical behavior is more complex and the aforementioned limits do not exist. If either p(x)(x - x_0) or q(x)(x - x_0)^2 fails to have a finite limit as x approaches x_0, the point is classified as irregular or essential.

Irregular singular points are noteworthy because the behavior of solutions near these points can be quite unpredictable and may involve terms that grow more rapidly than any polylogarithmic series can describe. Therefore, the solution techniques for irregular singular points often involve more sophisticated mathematical methods beyond the scope of power series solutions.

L'Hopital's Rule

Dealing with Indeterminate Forms

One tool that mathematicians frequently use to analyze limits involving indeterminate forms such as 0/0 and ∞/∞ is L'Hopital's Rule. This rule helps determine such limits by replacing the original indeterminate function with its derivatives.To apply L'Hopital's Rule, take the derivative of the numerator and the derivative of the denominator separately, then take the limit of this new fraction. It can be applied repeatedly until a determinate form is obtained or it becomes apparent that the limit is infinite. However, for L'Hopital's Rule to be valid, certain conditions must be met:

• The limit must initially be an indeterminate form.
• The derivatives of the numerator and denominator exist.
• The limit of the derivatives exists or tends toward infinity.

In the given exercise, L'Hopital's Rule is used to assess \[\lim_{x\to0}\frac{(\ln|x|)}{x}\] because the natural logarithm of zero is undefined, leading to an indeterminate form. This rule aids in revealing that the limit actually diverges, contributing to the classification of the point as an irregular singular point.

Natural Logarithm

Understanding the Logarithmic Function's Behavior

The natural logarithm, denoted as \(\ln(x)\), is a significant mathematical function characterized by its occurrence in many areas of mathematics and natural phenomena. It is the inverse of the exponential function with base \(e\), where \(e\) is Euler's number, approximately equal to 2.71828.

The function is defined only for positive real numbers, and as the input approaches zero, the logarithm goes to negative infinity, \(\ln(x) \to -\infty\) as \(x \to 0^+\). In the context of differential equations, the natural logarithm can introduce a singularity at \(x=0\), since it is not defined for \(x \leq 0\). The behavior near this singularity can significantly affect the classification of singular points, as shown in the exercise, where the natural logarithm contributes to the presence of an irregular singular point.