Problem 5
Question
Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ \left(x^{3}+4 x\right) y^{\prime \prime}-2 x y^{\prime}+6 y=0 $$
Step-by-Step Solution
Verified Answer
The singular point is at \( x = 0 \) and it is regular.
1Step 1: Identify the Differential Equation
The differential equation given is \( \left(x^3 + 4x\right) y'' - 2x y' + 6y = 0 \).
2Step 2: Rewrite in Standard Form
To identify singular points, first rewrite the differential equation in standard form: \[ y'' + P(x) y' + Q(x) y = 0 \]Divide the entire equation by \( x^3 + 4x \), the coefficient of \( y'' \):\[ y'' - \frac{2x}{x^3+4x} y' + \frac{6}{x^3+4x} y = 0 \]
3Step 3: Identify Singular Points
Singular points occur where the coefficients \( P(x) \) and \( Q(x) \) become undefined. This occurs when \( x^3 + 4x = 0 \). Factor the expression:\[ x(x^2 + 4) = 0 \]So the singular points are \( x = 0 \) and \( x = \pm 2i \). However, we focus on real points, so \( x = 0 \) is the singular point.
4Step 4: Determine If the Singular Point is Regular or Irregular
A singular point is regular if:\[ (x - x_0)P(x) \] and \[ (x - x_0)^2Q(x) \] are analytic at \( x_0 \).Substitute \( x_0 = 0 \):\( P(x) = \frac{-2x}{x^3+4x} = \frac{-2}{x^2+4} \)\( (x - 0)P(x) = \frac{-2x}{x^2+4} \) is analytic at \( x = 0 \). \( Q(x) = \frac{6}{x^3+4x} = \frac{6}{x(x^2+4)} \)\( (x - 0)^2Q(x) = \frac{6x^2}{x(x^2+4)} = \frac{6x}{x^2+4} \) is also analytic at \( x = 0 \).Since both expressions are analytic at \( x = 0 \), the singular point is regular.
Key Concepts
Regular Singular PointsIrregular Singular PointsOrdinary Differential EquationsAnalytic Functions
Regular Singular Points
Regular singular points hold an essential place in the study of differential equations. In ordinary differential equations, these are points where the equation has undefined coefficients, but not in a way that is too troublesome. Regular singular points might seem complex, but there's a way to identify them quite easily.
Consider a differential equation in the standard form: \[ y'' + P(x) y' + Q(x) y = 0 \]A singular point, \( x_0 \), is termed regular if:
In the given example, we have verified these conditions for \( x = 0 \). Both conditions hold true, making \( x = 0 \) a regular singular point.
Consider a differential equation in the standard form: \[ y'' + P(x) y' + Q(x) y = 0 \]A singular point, \( x_0 \), is termed regular if:
- The product \((x - x_0)P(x)\) is analytic at \( x_0 \).
- The product \((x - x_0)^2Q(x)\) is also analytic at \( x_0 \).
In the given example, we have verified these conditions for \( x = 0 \). Both conditions hold true, making \( x = 0 \) a regular singular point.
Irregular Singular Points
Sometimes, singular points do not follow the nice conditions of regularity. These points are known as irregular singular points. Unlike regular singular points, irregular ones can result in more complex and unpredictable solution behavior near the point.
The intricacy arises because the expressions \((x - x_0)P(x)\) or \((x - x_0)^2Q(x)\) fail to be analytic at \( x_0 \). In simple terms, this could mean abrupt changes or singularities that make solving and predicting solutions a complex task. Thankfully, in many real-life problems, such points are less frequent or can be transformed into a simpler form.
In the exercise, there are no irregular singular points identified because \( x = 0 \) is regular. But understanding irregular singular points is crucial for every student, as they encounter more complicated differential equations.
The intricacy arises because the expressions \((x - x_0)P(x)\) or \((x - x_0)^2Q(x)\) fail to be analytic at \( x_0 \). In simple terms, this could mean abrupt changes or singularities that make solving and predicting solutions a complex task. Thankfully, in many real-life problems, such points are less frequent or can be transformed into a simpler form.
In the exercise, there are no irregular singular points identified because \( x = 0 \) is regular. But understanding irregular singular points is crucial for every student, as they encounter more complicated differential equations.
Ordinary Differential Equations
Ordinary Differential Equations (ODEs) are a central part of mathematical studies, often used to model real-world phenomena. These equations relate functions to their derivatives and typically involve finding unknown functions from the given differential relations.
In standard form, an ODE looks like this: \[ y'' + P(x) y' + Q(x) y = 0 \]Here, \( y \) is the dependent variable, and \( x \) is the independent one. The terms \( P(x) \) and \( Q(x) \) can vary significantly, creating diverse scenarios for solution behaviors. Many problems in physics, engineering, and other sciences boil down to solving such equations.
In learning about singular points, students gain insight into special spots where the usual conditions are disrupted. It helps in developing clever methods to solve these equations, especially around those special points.
In standard form, an ODE looks like this: \[ y'' + P(x) y' + Q(x) y = 0 \]Here, \( y \) is the dependent variable, and \( x \) is the independent one. The terms \( P(x) \) and \( Q(x) \) can vary significantly, creating diverse scenarios for solution behaviors. Many problems in physics, engineering, and other sciences boil down to solving such equations.
In learning about singular points, students gain insight into special spots where the usual conditions are disrupted. It helps in developing clever methods to solve these equations, especially around those special points.
Analytic Functions
Analytic functions are foundational in understanding mathematical behavior because they exhibit smoothness and continuous derivatives of all orders around a point. For a function to be analytic at a point \( x_0 \), it should not only be defined there but also have a convergent power series in some neighborhood of that point.
This property makes analytic functions a favorite in mathematics due to their wide applicability and the ease with which they can be analyzed. When working with ODEs, especially at singular points, identifying whether expressions are analytic can simplify the task significantly.
In the context of singular points in differential equations, if the crucial expressions at a singular point are analytic, a regular singular point is identified. This makes finding solutions more manageable and provides predictability to the behavior of solutions near those points.
This property makes analytic functions a favorite in mathematics due to their wide applicability and the ease with which they can be analyzed. When working with ODEs, especially at singular points, identifying whether expressions are analytic can simplify the task significantly.
In the context of singular points in differential equations, if the crucial expressions at a singular point are analytic, a regular singular point is identified. This makes finding solutions more manageable and provides predictability to the behavior of solutions near those points.
Other exercises in this chapter
Problem 4
\(x=0\) is an ordinary point of a certain linear differential equation. After the assumed solution \(y=\sum_{n=0}^{\infty} c_{n} x^{n}\) is substituted into the
View solution Problem 4
Find the radius of convergence and interval of convergence for the given power series. $$ \sum_{k=0}^{\infty} k !(x-1)^{k} $$
View solution Problem 5
Suppose the powers series \(\sum_{n=0} c_{k}(x-4)^{k}\) is known to converge at \(-2\) and diverge at 13 . Discuss whether the series converges at \(-7,0,7,10\)
View solution Problem 6
Determine the singular points of the given differential equation. Classify each singular point as regular or irregular. $$ x^{2}(x-5)^{2} y^{\prime \prime}+4 x
View solution