Question

determine whether the given boundary value problem is self-adjoint. $$ y^{\prime \prime}+\lambda y=0, \quad y(0)=0, \quad y(\pi)+y^{\prime}(\pi)=0 $$

Short answer

In summary, the given boundary value problem is self-adjoint since its ODE is already in self-adjoint form and its boundary conditions are also in self-adjoint form.

Step-by-step solution

Transform the ODE to the self-adjoint form if needed.

The given ODE is already in the self-adjoint form: $$ \frac{d}{dx} \left[ 1 \cdot \frac{dy}{dx} \right] + 0 \cdot \frac{dy}{dx} + \lambda y = 0 \Rightarrow y'' + \lambda y = 0. $$

Check if the given boundary conditions are self-adjoint.

The given boundary conditions for the problem are: $$ y(0) = 0 \quad \text{and} \quad y(\pi) + y'(\pi) = 0. $$ These boundary conditions can be written as $$ 1 \cdot y(0) + 0 \cdot y'(0) = 0 \quad \text{and} \quad 1 \cdot y(\pi) + 1 \cdot y'(\pi) = 0. $$ They are both in the self-adjoint form: $$ \alpha_1 y(a) + \alpha_2 y'(a) = 0 \quad \text{and} \quad \beta_1 y(b) + \beta_2 y'(b) = 0. $$ Since the given ODE is already in self-adjoint form and the boundary conditions are also self-adjoint, the given boundary value problem is self-adjoint.

Key concepts

Ordinary Differential Equations

Ordinary Differential Equations (ODEs) are equations involving functions and their derivatives. Unlike partial differential equations, which involve multiple independent variables, ODEs typically involve just one. In the simplest form, ODEs like the one in our exercise represent mathematical relationships between varying quantities that can be expressed using derivatives.

ODEs are crucial in understanding natural phenomena and appear in fields such as physics, biology, and finance. They allow us to describe the behavior of dynamic systems. For example:

• The growth rate of a population in biology.
• The motion of objects in physics.
• The rate of change in financial markets.

In our exercise, we work with the equation \(y'' + \lambda y = 0\). Here, \(y''\) is the second derivative of \(y\) with respect to \(x\), indicating how the rate of change itself is changing. Solving such ODEs helps us predict and understand complex behaviors.

Boundary Conditions

Boundary conditions are additional constraints provided in differential equations to determine specific solutions from a general solution. These conditions are essential because many differential equations have infinitely many solutions, and boundary conditions help select the one that fits a particular physical scenario.

In the given problem, the boundary conditions are \(y(0) = 0\) and \(y(\pi) + y'(\pi) = 0\). These conditions specify the behavior of functions at the boundaries—\(x = 0\) and \(x = \pi\) in this case.

• First condition: \(y(0) = 0\) signifies that at \(x = 0\), the function \(y\) must be zero.
• Second condition: \(y(\pi) + y'(\pi) = 0\) imposes that at \(x = \pi\), the sum of the function and its derivative is zero.

Boundary conditions dictate the specific shape and nature of the solution curve across the interval, ensuring it adheres to these prescribed conditions at the endpoints.

Eigenvalue Problems

Eigenvalue problems in differential equations involve finding the characteristic values (eigenvalues) for which there are non-trivial solutions (eigenfunctions) to the equations. These problems are essential in mathematical physics and engineering because they often describe natural frequencies of systems, resonances, or stability contours.

In the equation \(y'' + \lambda y = 0\), \(\lambda\) plays the role of an eigenvalue. Solving such problems leads to a deeper understanding of systems such as vibrating strings or heat transfer domains.

• Eigenvalues, \(\lambda\), represent special parameter values where the system exhibits particular resonant behaviors.
• Finding the correct \(\lambda\) means determining which values lead to valid, permissible solutions under the given boundary conditions.

Eigenvalue problems often lead to quantization, where only specific values are possible, fundamentally linking the abstract mathematics to real-world phenomena. In our boundary value problem, determining whether it is self-adjoint is a step toward confirming that our approach will lead to legitimate eigenvalue solutions.