Question

Find the eigenvalues and eigenfunctions of the given boundary value problem. Assume that all eigenvalues are real. \(y^{\prime \prime}+\lambda y=0, \quad y^{\prime}(0)=0, \quad y^{\prime}(\pi)=0\)

Short answer

The eigenvalues are given by λ = \(n^2\), where n is an integer. The corresponding eigenfunctions are \(y_n(x) = A_n\sin(nx)\), where \(A_n\) denotes constants for each integer value of n.

Step-by-step solution

Identify the type of differential equation

The given differential equation is a linear, second-order, homogeneous differential equation. We have: \(y^{\prime\prime} + \lambda y = 0\) with boundary conditions: \(y^{\prime}(0) = 0\) \(y^{\prime}(\pi) = 0\) It's important to notice that λ, the eigenvalues, are still unknown.

Solve the differential equation for different λ cases

We need to consider 3 cases: when λ>0, λ<0, and λ=0. Let's analyze each one separately: 1. If λ > 0, then define µ = \(\sqrt{\lambda}\) and solve the differential equation: The general solution would be: \(y(x) = A\sin(\mu x) + B\cos(\mu x)\) 2. If λ < 0, then define µ = \(\sqrt{-\lambda}\) and solve the differential equation: The general solution would be: \(y(x) = A\sinh(\mu x) + B\cosh(\mu x)\) 3. If λ = 0, we are left with the original differential equation: \(y^{\prime\prime} = 0\) The general solution would be: \(y(x) = Ax + B\)

Apply the boundary conditions

Now, apply the given boundary conditions to the general solutions for each case. 1. λ > 0, \(y(x) = A\sin(\mu x) + B\cos(\mu x)\) \(y^{\prime}(0) = 0 \Rightarrow A\mu\sin(\mu * 0) + B(-\mu\cos(\mu * 0)) = 0 \Rightarrow -B\mu = 0\) Since \(\mu \neq 0\) in this case, we have \(B = 0\). Thus, \(y(x) = A\sin(\mu x)\) Now, apply the second boundary condition, \(y^{\prime}(\pi) = 0 \Rightarrow A\mu\sin(\mu\pi) = 0\) For this condition to hold, we need \(\sin(\mu\pi) = 0\) which implies \(\mu\pi = n\pi\) and \(\mu = n\) for integer values of n. So, λ = \(n^2\). Now, our eigenvalues are \(\lambda_n = n^2\) and eigenfunctions are: \(y_n(x) = A_n\sin(nx)\), where \(A_n\) denotes constants for each n. 2. λ < 0, \(y(x) = A\sinh(\mu x) + B\cosh(\mu x)\) Unfortunately, for this case, no eigenvalues and eigenfunctions can be derived from the given boundary conditions. 3. λ = 0, \(y(x) = Ax + B\) For the given boundary conditions, no eigenvalues and eigenfunctions can be derived from this case, as well.

Write the final result

Based on the two boundary conditions, we found the eigenvalues and their corresponding eigenfunctions only for the λ > 0 case. The eigenvalues are given by λ = \(n^2\), where n is an integer, and their corresponding eigenfunctions are: \(y_n(x) = A_n\sin(nx)\) where \(A_n\) denotes constants for each integer value of n.

Key concepts

Boundary Value Problem

A boundary value problem is a type of differential equation along with a set of constraints called boundary conditions. In the context of physical systems, these problems often arise when we are interested in finding a function that describes a system's behavior under certain fixed conditions. For example, in our exercise, we are dealing with the differential equation
\(y^{\prime \prime} + \lambda y = 0\)
with the boundary conditions \(y^{\prime}(0) = 0\) and \(y^{\prime}(\pi) = 0\). These conditions stipulate that the derivative of y, which could represent a physical property like velocity, must be zero at both endpoints of the interval \([0, \pi]\). Understanding boundary value problems is crucial, as they are relevant in fields such as physics, engineering, and applied mathematics.

Linear Second-Order Differential Equation

Linear second-order differential equations are of the form
\(a_2(x)y^{\prime\prime} + a_1(x)y^{\prime} + a_0(x)y = g(x)\)
where \(a_2\), \(a_1\), and \(a_0\) are coefficients that may vary with x, and \(g(x)\) is a known function called the nonhomogeneous term. When \(g(x)\) is zero, the equation is called homogeneous. The given exercise features a homogeneous linear second-order differential equation, which simplifies the process since these equations possess special properties that allow for the superposition of solutions—a fundamental concept when discussing eigenvalues and eigenfunctions.

Homogeneous Differential Equation

In our exercise, the homogeneous differential equation is expressed as
\(y^{\prime\prime} + \lambda y = 0\)
where 'homogeneous' implies that the right side of the equation is zero. This condition hints at a crucial characteristic: any multiple of a solution is also a solution, which leads to the idea of eigenfunctions—a set of functions that, when operated on by a differential operator, yield a scalar multiple (the eigenvalue) of the original function. Homogeneous equations are foundational in understanding the vibrational modes of a system, making them integral in physics and engineering.

Eigenvalue Problem Solving

The eigenvalue problem solving process is a method to find special values, known as eigenvalues, for which a differential equation has non-trivial solutions—the eigenfunctions. In the given exercise, the problem was to determine the eigenvalues \(\lambda\) and corresponding eigenfunctions for the boundary value problem.
The strategy starts with solving the differential equation for various cases depending on the sign of \(\lambda\). Once general solutions are found, boundary conditions are applied to filter out the eigenvalues and eigenfunctions.
Only for \(\lambda > 0\), we obtained meaningful solutions: eigenvalues \(\lambda_n = n^2\) and associated eigenfunctions \(y_n(x) = A_n\sin(nx)\), where \(n\) is an integer. This method is instrumental in systems where quantization of properties, such as the natural frequencies of a drumhead or the energy levels of a quantum system, occurs.