Question

In each of Problems 7 through 10 determine the form of the eigenfunctions and the determinantal equation satisfied by the nonzero eigenvalues. Determine whether \(\lambda=0\) is an eigenvalue, and find approximate values for \(\lambda_{1}\) and \(\lambda_{2},\) the nonzero eigenvalues of smallest absolute value. Estimate \(\lambda_{n}\) for large values of \(n\). Assume that all eigenvalues are real. $$ \begin{array}{l}{y^{\prime \prime}+\lambda y=0} \\ {y(0)=0, \quad y(\pi)+y^{\prime}(\pi)=0}\end{array} $$

Short answer

Question: Determine the eigenfunctions, the determinantal equation satisfied by the nonzero eigenvalues, and check if λ=0 is an eigenvalue. Find the approximate values for λ₁ and λ₂, the nonzero eigenvalues of the smallest absolute value. Estimate λₙ for large values of n, assuming that all eigenvalues are real. Answer: The eigenfunctions are given by \(y_n(x) = \sin{k_n x}\), where \(k_n = \frac{(2n+1)}{2}\) and n is a non-negative integer. The determinantal equations satisfied by the nonzero eigenvalues are given by \(λ_n = \frac{(2n+1)^2}{4}\). λ=0 is not an eigenvalue. The approximate values for λ₁ and λ₂ are \(\frac{9}{4}\) and \(\frac{25}{4}\), respectively. For large values of n, λₙ can be estimated as \(λ_n \approx n^2\).

Step-by-step solution

Find the general solution of the given differential equation

First, we need to find the general solution of the given differential equation \(y''+\lambda y=0\). This is a homogeneous linear second order ODE and has general solutions depending on the sign of λ: 1. If \(λ > 0\), then let \(λ = k^2\), where \(k > 0\). The general solution becomes \(y(x) = A\sin{kx} + B\cos{kx}\). 2. If \(λ = 0\), then the general solution is \(y(x) = Ax + B\). 3. If \(λ < 0\), then let \(λ = -k^2\), where \(k > 0\). The general solution becomes \(y(x) = Ae^{kx} + Be^{-kx}\).

Apply the boundary conditions

Next, we will apply the given boundary conditions \(y(0) = 0\) and \(y(\pi) + y'(\pi) = 0\). Case 1: \(λ > 0\) We start with the general solution \(y(x) = A\sin{kx} + B\cos{kx}\). Apply the first boundary condition \(y(0) = 0\). This gives us $$0 = A\sin{0} + B\cos{0} \Rightarrow B = 0.$$ Now, the solution is \(y(x) = A\sin{kx}\). Apply the second boundary condition \(y(\pi) + y'(\pi) = 0\). $$0 = A\sin{k\pi} + A{k}\cos{k\pi}$$ We notice that we need the \(A{k}\cos{k\pi}\) term because the \(\sin{k\pi}\) term will be zero as sine of any integral multiple of \(\pi\) is zero. Therefore, we must have \(A\neq 0\) and the \(k\) value should satisfy $$0 = A{k}\cos{k\pi}$$ Let \(\cos{k\pi} = -1\) for the even integer values of n such that \(k_n=\frac{(2n+1)}{2}\) with n a non-negative integer (n = 0, 1, 2,...). The eigenvalues are \(λ_n = k_n^2 = \frac{(2n+1)^2}{4}\). The corresponding eigenfunctions are \(y_n(x) = \sin{k_n x}\). Case 2: \(λ = 0\) We have the general solution \(y(x) = Ax + B\). Apply the first boundary condition \(y(0) = 0\), which gives us \(B = 0\). So, the solution is \(y(x) = Ax\). Apply the second boundary condition \(y(\pi) + y'(\pi) = 0\), which gives us \(0 = A\pi + A\). This equation has no nontrivial solutions for A, meaning λ=0 is not an eigenvalue. Case 3: \(λ < 0\) In this case, the solutions are exponentially growing or decaying functions which do not satisfy the given boundary conditions. Therefore, we don't need to investigate this case.

Answering the questions in the problem

We have the nonzero eigenvalues \(λ_n = \frac{(2n+1)^2}{4}\), the eigenfunctions \(y_n(x) = \sin{k_n x}\), and λ=0 is not an eigenvalue. The nonzero eigenvalues of the smallest absolute value are: $$λ_1 = \frac{(2(1)+1)^2}{4} = \frac{9}{4} \quad \text{and} \quad λ_2 = \frac{(2(2)+1)^2}{4} = \frac{25}{4}.$$ For a large value of \(n\), the eigenvalue \(λ_n\) can be estimated as: $$λ_n = \frac{(2n+1)^2}{4} \approx \frac{4n^2}{4} = n^2$$ In conclusion, we have determined the form of the eigenfunctions, the determinantal equation satisfied by the nonzero eigenvalues, checked that λ=0 is not an eigenvalue, and found approximate values for λ₁ and λ₂, as well as estimating λₙ for large values of n.

Key concepts

Boundary Value Problem

Boundary value problems are a type of differential equation problem where you are given additional conditions, known as boundary conditions, rather than initial conditions. This type of problem is crucial in physics and engineering, where conditions at the boundaries are often known, but not in the middle of the domain.
In the given problem, we are asked to solve the differential equation along with two boundary conditions:

• \(y(0) = 0\): The solution must equal zero when \(x = 0\).
• \(y(\pi) + y'(\pi) = 0\): The sum of the solution and its first derivative must equal zero when \(x = \pi\).

This requires us to find solutions that fit within these constraints, which may result in discrete solutions known as eigenvalues and their corresponding eigenfunctions.

Second Order Differential Equations

Second order differential equations, such as \(y'' + \lambda y = 0\), are equations that involve the second derivative of a function. These are pivotal in describing systems that depend on acceleration, curvature, or other second-derivative terms.
The general solution to these equations varies depending on the eigenvalue parameter \(\lambda\):

• If \(\lambda > 0\), the solution takes the form of trigonometric functions: \(y(x) = A\sin(kx) + B\cos(kx)\), where \(k = \sqrt{\lambda}\).
• If \(\lambda = 0\), the solution is linear: \(y(x) = Ax + B\).
• If \(\lambda < 0\), the solution involves exponential functions: \(y(x) = Ae^{kx} + Be^{-kx}\), where \(k = \sqrt{-\lambda}\).

It is this diversity in solution types that makes second order differential equations a rich area for exploration in advanced mathematics.

Eigenfunctions

An eigenfunction is a special type of function associated with a particular eigenvalue of a differential operator. They represent modes of vibration, states of a physical system, or natural modes, among other interpretations. For boundary value problems, these solutions satisfy both the differential equation and the boundary conditions.
In the context of the exercise, the goal is to find eigenfunctions that satisfy the conditions \(y(0) = 0\) and \(y(\pi) + y'(\pi) = 0\). Applying these boundary conditions to the general solutions leads to specific solutions consistent with these requirements. Such solutions are often sinusoidal when considering positive \(\lambda\). For example:

• For \( \lambda_n = \frac{(2n+1)^2}{4} \), the eigenfunctions are given by \(y_n(x) = \sin(k_n x)\) with \(k_n = \frac{(2n+1)}{2}\).

These functions are orthogonal and form a basis, allowing for the reconstruction of other functions within the same space using these eigenfunctions.

Homogeneous Linear ODE

A homogeneous linear ordinary differential equation (ODE) is one where the function and its derivatives are linearly related and include no terms without the function or its derivatives (i.e., no constant terms). The motto of homogeneity means that the principle of superposition applies: the sum of solutions is also a solution.
In the equation \(y'' + \lambda y = 0\), this ODE is homogeneous since the terms involve only the function \(y\) and its second derivative. The term "linear" refers to the fact that each instance of \(y\) and its derivatives are to the first power.
Solving homogeneous linear ODEs, especially with boundary conditions, often involves looking for characteristic-style solutions (e.g., exponential or trigonometric) and applying relevant conditions at the boundaries to find those modes or eigenfunctions that satisfy both the equation and its requisite conditions.