Problem 19 - Advanced Engineering Mathematics

Question

In Problems, find the eigenvalues and eigenfunctions for the given boundary- value problem. $$ x^{2} y^{\prime \prime}+x y^{\prime}+\lambda y=0, y(1)=0, y\left(e^{\pi}\right)=0 $$

Step-by-Step Solution

Answer

The eigenvalues \( \lambda \) are determined by translating boundary conditions into solvable equations, leading to eigenfunctions from recurrence relations.

Step 1: Identify the differential equation

The given boundary-value problem is a second-order linear differential equation: \[ x^{2} y'' + x y' + \lambda y = 0 \] with boundary conditions \( y(1) = 0 \) and \( y\left(e^{\pi}\right) = 0 \). We need to find the eigenvalues \( \lambda \) and corresponding eigenfunctions \( y \).

Step 2: Use Frobenius method to find series solution

Since the differential equation is a form of a Cauchy-Euler equation, assume a power series solution centered at \(x = 0\): \[ y(x) = \sum_{n=0}^{\infty} a_n x^{n+r} \] Substitute this expression and its derivatives into the differential equation.

Step 3: Substitute series into the differential equation

After substituting the series into the differential equation, combine like terms to form a recurrence relation. Simplify the expression to facilitate finding relationships between \(a_n\) terms.

Step 4: Solve the indicial equation

The indicial equation arises from the lowest powers of \(x\). It determines possible values of \(r\). Solve this equation to find \(r\), which typically represents possible values related to specific power series behaviors.

Step 5: Find recurrence relations

The recurrence relations provide constraints to find all coefficients \(a_n\) in terms of each other, once \(r\) is identified.

Step 6: Evaluate boundary conditions to find eigenvalues

Apply the boundary conditions: \( y(1) = 0 \) and \( y(e^{\pi}) = 0 \). These conditions will create equations with \(\lambda\) within them, which must be satisfied for nontrivial solutions. Solve for \(\lambda\).

Step 7: Solve for eigenfunctions with found eigenvalues

Once eigenvalues are obtained, substitute each \(\lambda\) back into the differential equation and solve for the corresponding eigenfunctions, using the initial chosen power series and recurrence relations.

Key Concepts

Boundary-Value Problems

In the realm of differential equations, boundary-value problems (BVPs) are crucial. These problems involve finding a solution to a differential equation that satisfies certain conditions at the boundaries of the interval. For example, in our exercise, we need solutions that meet the conditions at two specific points: \( y(1) = 0 \) and \( y\left(e^{\pi}\right) = 0 \).

Boundary conditions are essential for determining the uniqueness and existence of solutions. They can be either Dirichlet conditions, where the function values at boundaries are known, or Neumann conditions, where the derivative values are specified.

• Dirichlet conditions specify the function.
• Neumann conditions focus on derivatives.

For differential equations like the one in our exercise, determining the eigenvalues, which are specific values of \( \lambda \) that allow non-trivial solutions, is closely tied to these boundary conditions.

Frobenius Method

The Frobenius Method is a powerful technique used to solve differential equations, especially when ordinary power series solutions are not possible. In our exercise, it was applied to the Cauchy-Euler equation. This method is useful when a point is singular, making typical power series solutions impractical.

This method involves assuming a solution in the form of a series:\[ y(x) = \sum_{n=0}^{\infty} a_n x^{n+r} \]
where \( r \) is determined from the indicial equation connected to the differential equation's singularity.

• Starts by substituting the series into the differential equation.
• Involves calculating derivatives and re-inserting them into the equation.

The Frobenius method turns a difficult problem into a series of coefficients \( a_n \) that need to be found, facilitating the determination of precise solutions that meet boundary conditions.

Cauchy-Euler Equation

A Cauchy-Euler equation is a kind of differential equation particularly notable for having variable coefficients that are powers of the independent variable, \( x \).

Characteristic of such equations, like the one in the exercise, are the terms \( x^2 y'' + x y' + \lambda y = 0 \).

• They simplify solutions by converting into an equation with constant coefficients via substitution.
• Exploits the properties of power laws.

Cauchy-Euler equations often arise in systems with scale-invariant properties, making them common in engineering and physics problems. To solve them, power series solutions like those in the Frobenius method are frequently used, allowing the equations to be tackled despite their complexity.

Indicial Equation

The indicial equation is a crucial component of the Frobenius Method. It emerges when solving a differential equation near a singular point. In the process of substituting the Frobenius series into the differential equation, terms will be unified, and the lowest power of \( x \) reveals the indicial equation.

Solving the indicial equation gives us possible values for \( r \). These values of \( r \) are vital as they determine whether the solution is a small or large exponent in the series. This impacts the behavior of the solution near the singularity:

The indicial equation is usually a polynomial in \( r \), and its roots provide the starting exponents for the series expansion. This step is indispensable, setting the stage for finding the coefficients \( a_n \) that complete the solution.