Problem 11
Question
Put the given differential equation into the form (3) for each regular singular point of the equation. Identify the functions \(p(x)\) and \(q(x)\). $$ \left(x^{2}-1\right) y^{\prime \prime}+5(x+1) y^{\prime}+\left(x^{2}-x\right) y=0 $$
Step-by-Step Solution
Verified Answer
The functions are \(p(x) = \frac{5(x+1)}{x^2-1}\) and \(q(x) = \frac{x^2-x}{x^2-1}\).
1Step 1: Identify Singular Points
Identify where the given differential equation \[ \left(x^{2}-1\right) y^{\prime \prime}+5(x+1) y^{\prime}+\left(x^{2}-x\right)y=0 \]becomes singular. A singular point occurs where the coefficient of the highest derivative, \(x^2 - 1\), equals zero.Setting \(x^2 - 1 = 0\), we find: \[x^2 = 1\text{ hence } x = \pm 1.\]These points, \(x=1\) and \(x=-1\), are the singular points of the equation.
2Step 2: Transform Differential Equation to Standard Form (Optional)
To identify \(p(x)\) and \(q(x)\) in the equation, consider the form:\[ y^{\prime\prime} + p(x) y^{\prime} + q(x) y = 0 \]Rewrite the differential equation by dividing through by \(x^2 - 1\):\[ y^{\prime\prime} + \frac{5(x+1)}{x^2-1} y^{\prime} + \frac{x^2-x}{x^2-1} y = 0 \]This makes it clear that \(p(x) = \frac{5(x+1)}{x^2-1}\) and \(q(x) = \frac{x^2-x}{x^2-1}\).
3Step 3: Determine Regularity of Singular Points
A singular point \(x_0\) is regular if \((x-x_0)p(x)\) and \((x-x_0)^2q(x)\) are analytic at \(x_0\). Evaluate at both \(x = 1\) and \(x = -1\):For \(x = 1\): - \((x-1)p(x) = \frac{5(x+1)}{x+1} = 5\), is analytic at \(x_0=1\). - \((x-1)^2 q(x) = \frac{x^2-x}{x+1}\), is also analytic.For \(x = -1\): - \((x+1)p(x) = \frac{5(x+1)}{x-1} = \frac{5(x+1)}{x-1}\), appears to be indeterminate but can be analyzed further. - \((x+1)^2 q(x) = (x+1)\), analytic at \(x_0=-1\).Both \(x=1\) and \(x=-1\) are regular singular points.
Key Concepts
Differential EquationsSingular PointsAnalytic Functions
Differential Equations
A differential equation is a mathematical equation that relates some function with its derivatives. In simple terms, these equations involve rates of change. These types of equations appear in various areas such as physics, engineering, and economics.
Differential equations can be classified based on order, linearity, and homogeneity, among others. The order of a differential equation is defined by the highest derivative it contains. In our case, the given equation is a second-order differential equation because the highest derivative is the second derivative, denoted by \( y'' \).
Differential equations can be incredibly powerful tools for modeling real-world phenomena. Understanding how to manipulate and solve them provides insight into the dynamics of systems we wish to study.
Differential equations can be classified based on order, linearity, and homogeneity, among others. The order of a differential equation is defined by the highest derivative it contains. In our case, the given equation is a second-order differential equation because the highest derivative is the second derivative, denoted by \( y'' \).
Differential equations can be incredibly powerful tools for modeling real-world phenomena. Understanding how to manipulate and solve them provides insight into the dynamics of systems we wish to study.
Singular Points
Singular points in differential equations are critical as they indicate where solutions may become undefined or behave irregularly. These points occur where the coefficient of the highest derivative is zero. Understanding singular points is crucial for analyzing the behavior of solutions near these regions.
For the provided equation, singular points occur when \(x^2 - 1 = 0\), which leads to \(x = \pm 1\). At these points, the differential equation is not regular if the coefficients of the derivatives change drastically as \(x\) approaches these values. However, if certain conditions are met, these can still be regular singular points.
Regular singular points allow us to find solutions using methods like Frobenius series expansion. Evaluating the regularity of singular points helps determine the best approach for solving a differential equation.
For the provided equation, singular points occur when \(x^2 - 1 = 0\), which leads to \(x = \pm 1\). At these points, the differential equation is not regular if the coefficients of the derivatives change drastically as \(x\) approaches these values. However, if certain conditions are met, these can still be regular singular points.
Regular singular points allow us to find solutions using methods like Frobenius series expansion. Evaluating the regularity of singular points helps determine the best approach for solving a differential equation.
Analytic Functions
An analytic function is a function that is locally represented by a convergent power series around a point. Analytic functions are particularly important when working with differential equations at singular points.
In the context of the provided problem, the solutions around regular singular points depend on whether certain transformed functions are analytic. Specifically, expressions like \((x-x_0)p(x)\) and \((x-x_0)^2 q(x)\) must be evaluated for analyticity at each singular point \(x_0\).
For example, at \(x = 1\), the expression \((x-1)p(x) = 5\) is analytic because it simplifies to a constant value. Similarly, analyzing these expressions helps confirm the regularity of the singular points and ensures that a solution can be structured evenly around these points using series expansions.
In the context of the provided problem, the solutions around regular singular points depend on whether certain transformed functions are analytic. Specifically, expressions like \((x-x_0)p(x)\) and \((x-x_0)^2 q(x)\) must be evaluated for analyticity at each singular point \(x_0\).
For example, at \(x = 1\), the expression \((x-1)p(x) = 5\) is analytic because it simplifies to a constant value. Similarly, analyzing these expressions helps confirm the regularity of the singular points and ensures that a solution can be structured evenly around these points using series expansions.
Other exercises in this chapter
Problem 10
Use an appropriate infinite series method about \(x=0\) to find two solutions of the given differential equation. $$ y^{\prime \prime}-x y^{\prime}-y=0 $$
View solution Problem 10
Rewrite the given power series so that its general term involves \(x^{k}\). $$ \sum_{n=3}^{\infty}(2 n-1) c_{n} x^{n-3} $$
View solution Problem 11
In Problems, use the indicated change of variable to find the general solution of the given differential equation on the interval \((0, \infty)\). $$ x^{2} y^{\
View solution Problem 11
Rewrite the given expression as a single power series whose general term involves \(x^{k}\). $$ \sum_{\substack{n=1 \\ \infty}}^{\infty} 2 n c_{n} x^{n-1}+\sum_
View solution