Question

Use the method of Frobenius to find solutions near \(x=0\) of each of the differential equations in Exercises, $$ x^{2} y^{\prime \prime}-x y^{\prime}+\left(2 x^{2}+\frac{5}{9}\right) y=0 $$

Short answer

Using the method of Frobenius, we first assume a power series solution \(y(x) = \sum_{n=0}^{\infty} a_nx^{n + r}\), where \(a_n\) are the coefficients, and r is a constant called the index. We calculate the first and second derivatives, plug them into the given differential equation, and equate the powers of x. This gives us a recurrence relation for the coefficients: \(a_{n+1} = \frac{(n+r-d)(n+r+1)}{(n+1)(n+r)} a_n\), where d is a constant from the indicial equation \((n+r)(n+r-1) = d\). By analyzing the coefficients from this recurrence relation, we can find two linearly independent solutions near x=0, combining them to form the general solution for the given differential equation.

Step-by-step solution

Assumption of a Power Series Solution

Using the method of Frobenius, assume a solution in the form of a power series: \[y(x) = \sum_{n=0}^{\infty} a_nx^{n + r}\] where \(a_n\) are the coefficients, and r is a constant called the index.

Calculate the First and Second Derivative

Calculate the first and second derivatives of our assumed solution to plug them into the given differential equation. First derivative: \[y'(x) = \sum_{n=0}^{\infty} (n+r)a_nx^{n+r-1}\] Second derivative: \[y''(x) = \sum_{n=0}^{\infty} (n+r)(n+r-1)a_nx^{n+r-2}\]

Substitute into the Differential Equation

Substitute our assumed solution and its derivatives into the given differential equation: \[x^2 \sum_{n=0}^{\infty} (n+r)(n+r-1)a_nx^{n+r-2} - x \sum_{n=0}^{\infty} (n+r)a_nx^{n+r-1} + \left(2x^2 + \frac{5}{9}\right) \sum_{n=0}^{\infty} a_nx^{n+r} = 0\]

Equate the Powers of x and Simplify

Simplify the equation, equating the powers of x: \[\sum_{n=0}^{\infty} ((n+r)(n+r-1)a_nx^{n+r} - (n+r)a_nx^{n+r} + 2a_nx^{n+r} + \frac{5}{9}a_nx^{n+r}) = 0\] Cancel x terms and collect common terms: \[\sum_{n=0}^{\infty} a_nx^{n+r}((n+r)(n+r-1) - (n+r) + 2 + \frac{5}{9}) = 0\]

Set the Bracketed Terms to Zero

To have a non-trivial solution, we set the bracketed terms to zero: \[(n+r)(n+r-1) - (n+r) + 2 + \frac{5}{9} = 0\] Solve for r and set the indicial equation: \[(n+r)(n+r-1) = d\] where d is a constant.

Find Recurrence Relation

The coefficients \(a_n\) terms must satisfy the recurrence relation formula: \[a_{n+1} = \frac{(n+r-d)(n+r+1)}{(n+1)(n+r)} a_n\]

Analyze the Behavior Near x=0

We have found the power series solution to our differential equation. To analyze its behavior near x=0, we examine the coefficients generated by our recurrence relation formula. It is important to remember that the coefficients depend on the choice of r and the constant d, which can vary. The method of Frobenius allows us to find two linearly independent solutions near x=0, which can be combined to form a general solution. Hence, this method provides us with a way to discover solutions near x=0 for the given differential equation.

Key concepts

Power Series Solution

In mathematics, particularly when dealing with differential equations, it's quite common to seek solutions in the form of series where coefficients multiply powers of a variable. This technique is termed as seeking a "Power Series Solution.” The main idea behind a power series solution is to represent the unknown function as an infinite sum, where each term is a power of the independent variable. For the differential equation in this context, we assume the solution can be expressed as:\[ y(x) = \sum_{n=0}^{\infty} a_nx^{n+r} \]Here, \( a_n \) are coefficients that need to be determined, and \( r \) is a constant known as the index.
The beauty of the power series solution is its flexibility; it can be used to solve many types of differential equations, especially when they are challenging to solve with standard techniques.

Differential Equation

A differential equation is a mathematical equation involving derivatives of an unknown function. The goal is to find this function that fits the equation. In the given exercise, our differential equation is:\[ x^{2} y'' - x y' + \left(2x^{2} + \frac{5}{9}\right) y = 0 \]This is a second-order linear differential equation. Each term involves the unknown function \( y \) or its derivatives. The differential equation provided is homogeneous, meaning it is set equal to zero. Solving such equations often involves techniques like the Frobenius method or power series, especially near irregular points like \( x=0 \).
Here, the aim is to express the solution as a series expansion, making it easier to handle complex terms.

Recurrence Relation

In mathematics, a recurrence relation is an equation that recursively defines a sequence. In our context, it determines the coefficients of the power series solution.Once we substitute the power series and its derivatives into the differential equation, we obtain an expression in terms of\( x^n \). Setting the coefficients of these powers to zero gives us a recurrence relation:\[ a_{n+1} = \frac{(n+r-d)(n+r+1)}{(n+1)(n+r)} a_n \]This formula determines how each coefficient \( a_{n+1} \) is related to its preceding coefficient \( a_n \), helping construct the entire series solution. Such relations are crucial as they often provide insights into the behavior of solutions, especially as \( n \) increases.

Indicial Equation

The indicial equation is a crucial part when seeking a Frobenius series solution for a differential equation. It originates from setting the leading coefficients (those with the smallest power of the variable) to zero. In this problem, it's derived from simplifying and setting the lowest power terms’ coefficients to zero. This step usually involves terms containing the constant \( r \). By equating this to zero, an equation emerges called the indicial equation:\[(n+r)(n+r-1) + additional terms = 0\]This equation is vital as solving it determines the value of \( r \), the index. The solutions of the indicial equation provide information on the valid range of power series solutions. Values of \( r \) found are used to evaluate further recurrence relations and thus, construct possible series solutions.