Question

For the boundary conditions given below, obtain a functional \(\Lambda(y)\) whose stationary values give the eigenvalues of the equation $$ (1+x) \frac{d^{2} y}{d x^{2}}+(2+x) \frac{d y}{d x}+\lambda y=0, \quad y(0)=0, y^{\prime}(2)=0 $$

Short answer

The functional \Lambda(y) is \[\begin{equation}\Lambda(y) = \frac{1}{2} \int_0^2 \big((1+x) (\frac{dy}{dx})^2 - \beta y^2\big) \ dx \end{equation}\].

Step-by-step solution

- Write the given differential equation

The given differential equation is \[\begin{equation}(1+x) \frac{d^{2} y}{d x^{2}}+(2+x) \frac{d y}{d x}+\beta y=0\end{equation}\]where \beta \text{ is the parameter we want the eigenvalues for}.

- Apply the boundary conditions

The boundary conditions are \[\begin{equation} y(0) = 0 \end{equation}\]\[\begin{equation} y'(2) = 0 \end{equation}\].

- Rewrite the differential equation in self-adjoint form

The given differential equation can be written in self-adjoint form as \[\begin{equation} \frac{d}{dx} \big((1+x) \frac{d y}{d x}\big) + (2+x) \frac{d y}{d x} + \beta y = 0 \end{equation}\].

- Identify the functional

To find the functional \Lambda(y)\, we use the given boundary conditions and self-adjoint form. The functional can be written as \[\begin{equation} \Lambda(y) = \frac{1}{2} \int_0^2 \big((1+x) (\frac{dy}{dx})^2 - \beta y^2 \big) \ dx + \bigg( (1+x) y\frac{dy}{dx} \bigg)_{0}^{2} \end{equation}\].

- Simplify the functional

Simplify the term \bigg( (1+x) y\frac{dy}{dx} \bigg)_{0}^{2} \ using the boundary conditions. Since \ y(0)=0 \ and \ y'(2)=0 \:\[\begin{equation} \bigg( (1+x) y \frac{d y}{d x} \bigg)_{0}^{2} = 0 \end{equation}\]. Thus, the functional becomes \[\begin{equation}\Lambda(y) = \frac{1}{2} \int_0^2 \big((1+x) (\frac{dy}{dx})^2 - \beta y^2\big) \ dx \end{equation}\].

Key concepts

eigenvalues

In functional analysis, eigenvalues are special numbers associated with a differential equation. These numbers tell us how a function behaves under a given operation, like differentiation. When solving differential equations, the goal is to find these eigenvalues, which help in understanding the solutions.

For the given differential equation \[ (1+x) \frac{d^{2} y}{d x^{2}}+(2+x) \frac{d y}{d x}+\beta y=0 \]our task is to find the parameter \( \beta \) that serves as an eigenvalue. Finding these values gives insight into the particular solutions of the equation.

When boundary conditions like \( y(0) = 0 \) and \( y'(2) = 0 \) are applied, it becomes easier to solve for \( \beta \). The key is to identify how the function \( y(x) \) behaves with these constraints. Eigenvalues are critical in many areas, such as quantum mechanics, vibration analysis, and stability studies.

self-adjoint differential equation

A self-adjoint differential equation is one where the differential operator is equal to its adjoint. This property makes them easier to work with, especially when using variational methods.

To transform the given differential equation into a self-adjoint form, we rewrite it as:

\[ \frac{d}{dx} \big((1+x) \frac{d y}{d x}\big) + (2+x) \frac{d y}{d x} + \beta y = 0 \]

This form allows us to apply powerful mathematical tools to find solutions. Self-adjoint equations often have real eigenvalues and orthogonal eigenfunctions, which simplifies the analysis.

This transformation lets us leverage existing theorems and techniques, making it a fundamental step in solving differential equations.

boundary conditions

Boundary conditions are additional requirements that a solution to a differential equation must satisfy. They are essential in defining a unique solution for the equation. In the given problem, the boundary conditions are:

\[ y(0) = 0 \]\[ y'(2) = 0 \]

These conditions specify how the function \( y(x) \) behaves at particular points. For example, \( y(0) = 0 \) means that the function value at \( x=0 \) must be zero. Similarly, \( y'(2) = 0 \) means that the derivative of the function at \( x=2 \) should be zero.

By incorporating these conditions, we ensure that the function respects the physical or geometric constraints of the problem. This information is crucial in defining and solving the problem within a specific context.

variational methods

Variational methods are techniques used to find the functions that minimize or maximize a functional. A functional is an expression that depends on a function and its derivatives. In this exercise, the functional we are interested in is:

\[ \Lambda(y) = \frac{1}{2} \int_0^2 \big((1+x) (\frac{dy}{dx})^2 - \beta y^2\big) \, dx \]

By finding the stationary values of this functional, we obtain the eigenvalues. Stationary values occur where small changes in the function \( y(x) \) lead to no change in \( \Lambda(y) \). This is analogous to finding the peaks (maximum) and valleys (minimum) in a landscape.

Variational methods are powerful because they convert the problem of solving differential equations into an optimization problem. This is often easier to handle, especially using calculus of variations. It provides a systematic way to derive the equations and boundary conditions that a function must satisfy to optimize the functional.

Using variational methods in the given problem lets us harness these benefits, simplifying the task of finding the eigenvalues of the differential equation.