Question

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

Short answer

Question: Determine the type of the singular point for the given second-order differential equation: \((x^2)y^{\prime\prime} + 2x(e^x-1)y^{\prime} + e^{-x}\cos{x}y = 0\) Answer: The singular point at \(x=0\) is a regular singular point.

Step-by-step solution

Rewrite the equation in the standard form

The standard form of a second-order linear differential equation is given by: \begin{equation} y^{\prime\prime} + P(x)y^{\prime} + Q(x)y = 0 \end{equation} First, we need to rewrite the given equation in this form by dividing by \(x^2\): \begin{equation} y^{\prime\prime} + \frac{2(e^x-1)}{x^2}y^{\prime} + \frac{e^{-x}\cos{x}}{x^2}y = 0 \end{equation} Now, we have the equation in the standard form with \(P(x) = \frac{2(e^x-1)}{x^2}\) and \(Q(x) = \frac{e^{-x}\cos{x}}{x^2}\).

Find the singular points

To find the singular points, we need to identify the values of \(x\) for which either \(P(x)\) or \(Q(x)\) become infinite or undefined. By examining the two functions, we can see that both functions have potential singular points only at \(x=0\), as the denominators for both functions involve \(x^2\). So, \(x=0\) is the only candidate for a singular point in our equation.

Determine if the singular point is regular or irregular

Now, we need to determine if the singular point at \(x=0\) is regular or irregular. A singular point is regular if the following conditions hold for the coefficients \(P(x)\) and \(Q(x)\) near the singular point: 1) \((x-x_0)P(x)\) has a finite limit as \(x\to x_0\) 2) \((x-x_0)^2 Q(x)\) has a finite limit as \(x\to x_0\) Here, \(x_0 = 0\) is our singular point. We need to analyze the behavior of the two expressions above as \(x\to 0\): 1) As \(x\to 0\), \((x-0) \cdot \frac{2(e^x-1)}{x^2}=\frac{2(e^x-1)}{x}\). By applying L'Hôpital's rule to this expression, we get \(\lim_{x\to 0}\frac{2e^x}{1}=2\). So, the limit is finite. 2) As \(x\to 0\), \((x-0)^2 \cdot \frac{e^{-x}\cos{x}}{x^2}=e^{-x}\cos{x}\). Since both the exponential function and cosine function are finite and continuous at \(x=0\), their product is also finite as \(x\to 0\). Since both conditions are satisfied, we can conclude that the singular point at \(x=0\) is a regular singular point.

Summary

We rewrote the given equation in the standard form and identified the singular point at \(x=0\). By analyzing the behavior of the coefficients \(P(x)\) and \(Q(x)\) near the singular point, we determined that this point is a regular singular point.

Key concepts

Differential Equations

Differential equations are mathematical equations that relate a function to its derivatives. They play a significant role in modeling various physical phenomena, like motion, heat, electricity, and more.
Differential equations can be classified based on their order and linearity:

• Order: Determined by the highest derivative present in the equation. For example, a first-order differential equation involves only the first derivative.
• Linearity: A linear differential equation is one in which the function and its derivatives appear to the power of one (i.e., they are not multiplied or divided by each other).

These equations can often be complex. Understanding how to identify and work with singular points, especially in linear differential equations, is crucial for finding solutions that reflect real-world scenarios.

Regular Singular Point

A regular singular point is a specific type of singularity in a differential equation where, even though the functions involved may become infinite or undefined, a kind of "controlled behavior" or pattern still exists.
To determine if a point is a regular singular point in a differential equation:

• Consider the functions derived from the standard form of the equation.
• Check if certain conditions hold near the singular point using the coefficients of the equation.

Condition Checking

These conditions involve limits:

• (x-x_0) imes P(x) should approach a finite limit as x o x_0.
• (x-x_0)^2 imes Q(x) should also approach a finite limit as x o x_0.

Understanding these conditions helps in classifying the singular points and aids in finding solutions, especially when using power series methods.

Second-order Linear Differential Equation

A second-order linear differential equation includes the second derivative of a function. It follows a standard format that assists in understanding its components and finding solutions:

• The standard form is: \[ y'' + P(x)y' + Q(x)y = 0 \]
• Here, y'' represents the second derivative of the function, and P(x) and Q(x) are coefficient functions of x.

This type of equation is crucial as it often models physical systems in engineering, physics, and other sciences.

In practice, one rewrites given equations to this standard form. This allows more straightforward analysis of their behaviors and singularities, enabling the application of various solution methods, such as series expansions or numerical techniques.

Singularities in Differential Equations

Singularities in differential equations refer to points where the equation ceases to be well-behaved, such as where the coefficients become infinite or undefined. Recognizing and understanding these points is pivotal for solving these equations.
Types of singularities:

• Regular Singular Points: Points where the conditions allow for solutions to be expanded in a series, and certain limits are finite.
• Irregular Singular Points: Points that do not meet the criteria for regular singular points and often lead to more complex solution behaviors.

Importance of Identifying Singularities

Identifying the type of singularity helps in choosing the correct method for finding solutions. This is particularly useful in power series solutions, where the behavior at singular points determines the applicability of such methods.
By understanding these concepts, students can analyze differential equations more deeply, predict the behavior of solutions, and apply appropriate techniques accordingly.