Problem 11 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

Eigenvalues: \( \lambda_n = n^2 \); Eigenfunctions: \( y_n(x) = \sin(nx) \) for \( n = 1, 2, 3, \ldots \).

Step 1: Understanding the Problem

We are tasked with finding the eigenvalues and eigenfunctions for the differential equation \( y'' + \lambda y = 0 \) subject to boundary conditions \( y(0) = 0 \) and \( y(\pi) = 0 \). This is a standard Sturm-Liouville problem.

Step 2: Solving the Differential Equation

The general solution of the differential equation \( y'' + \lambda y = 0 \) depends on the sign of \( \lambda \). There are three cases: \( \lambda = 0 \), \( \lambda > 0 \), and \( \lambda < 0 \). For \( \lambda = 0 \), the solution is \( y(x) = C_1 + C_2x \); for \( \lambda > 0 \), let \( \lambda = \mu^2 \), the solution is \( y(x) = A \cos(\mu x) + B \sin(\mu x) \); for \( \lambda < 0 \), let \( \lambda = -\mu^2 \), the solution is \( y(x) = A \cosh(\mu x) + B \sinh(\mu x) \).

Step 3: Analyzing Eigenvalues: Case \( \lambda = 0 \)

For \( \lambda = 0 \), the solution \( y(x) = C_1 + C_2x \) leads to \( y(0) = C_1 = 0 \). The condition \( y(\pi) = 0 \) implies \( C_2 \pi = 0 \), so \( C_2 = 0 \). Thus, there is no non-trivial solution for \( \lambda = 0 \).

Step 4: Analyzing Eigenvalues: Case \( \lambda > 0 \)

For \( \lambda = \mu^2 > 0 \), the solution is \( y(x) = A \cos(\mu x) + B \sin(\mu x) \). Applying \( y(0) = 0 \) gives \( A = 0 \), so \( y(x) = B \sin(\mu x) \). Applying \( y(\pi) = 0 \) results in \( B \sin(\mu \pi) = 0 \). For a non-trivial solution, \( \sin(\mu \pi) = 0 \), implying \( \mu \pi = n\pi \) where \( n \) is an integer, so \( \mu = n \). Therefore, \( \lambda = n^2 \).

Step 5: Analyzing Eigenvalues: Case \( \lambda < 0 \)

For \( \lambda = -\mu^2 < 0 \), the solution is \( y(x) = A \cosh(\mu x) + B \sinh(\mu x) \). Boundary conditions \( y(0) = 0 \) imply \( A = 0 \), so \( y(x) = B \sinh(\mu x) \). Applying \( y(\pi) = 0 \) results in \( B \sinh(\mu \pi) = 0 \), which only holds for \( B = 0 \) since \( \sinh(\mu \pi) eq 0 \). Thus, there is no non-trivial solution for \( \lambda < 0 \).

Step 6: Finding the Eigenfunctions

From the non-trivial solutions for \( \lambda = n^2 \) where \( n \) is an integer, the eigenfunctions are \( y_n(x) = B \sin(nx) \) with \( n = 1, 2, 3, \ldots \) and \( B eq 0 \). Typically, \( B \) is chosen for convenience, often \( B = 1 \), giving eigenfunctions \( y_n(x) = \sin(nx) \).

Key Concepts

Sturm-Liouville theory

The problem you're dealing with falls under the umbrella of Sturm-Liouville theory. This theory is fundamentally about solving a special type of differential equation called the Sturm-Liouville differential equation. It usually comes with certain conditions, known as boundary conditions, that solutions must satisfy. Here's why it's special:

• It applies to linear second-order differential equations.
• The solutions, called eigenfunctions, provide a basis for many types of functions, much like how sine and cosine functions do for periodic functions.
• The parameters you're finding, known as eigenvalues, play important roles in various physical systems.

In simpler terms, this theory helps us understand how complex systems behave—like how a guitar string vibrates. Applying this theory to your problem involves finding the right frequencies (eigenvalues) and shapes (eigenfunctions) of vibration that satisfy the given conditions.

Boundary-value problems

When working with boundary-value problems, you're dealing with conditions at two different points. For your differential equation, these conditions are at points 0 and \( \pi \). This means you're looking for a function that not only satisfies the differential equation but also meets the given conditions at these boundary points. In practice, here's what this involves:

• Solving the differential equation itself.
• Applying the boundary conditions to find which solutions are valid (or non-trivial).
• Ensuring that the solution's behavior at the boundaries is precisely what's required.

The boundary-value problem is critical for finding meaningful solutions, especially in physics and engineering where it translates to meeting specific practical needs—like ensuring a bridge doesn't collapse or a beam doesn't bend unexpectedly.

Differential equations

Differential equations, such as the one you're working on, \( y'' + \lambda y = 0 \), form the backbone of many scientific disciplines. They describe how quantities change and interact. A second-order differential equation like this one involves the second derivative, telling us about the acceleration or concavity of a function. In the context of your exercise, consider these points:

• The equation's specific form captures the balance between the curve of a function and how it scales with a parameter \( \lambda \).
• The goal is to find solutions where the function remains zero at specified points—at 0 and \( \pi \).
• Solutions vary depending on whether \( \lambda \) is positive, negative, or zero, which leads to different types of functions—trigonometric, exponential, etc.

Understanding differential equations is crucial as they virtually appear everywhere in modeling real-world phenomena—from the growth of populations to the movement of particles.

Eigenfunctions

Eigenfunctions are the special solutions to differential equations within the framework of boundary-value problems. In your case, they are specific forms that "fit" the equation \( y'' + \lambda y = 0 \) while also obeying your boundary conditions. Here's a simple way to see them:

• They are the shapes (or modes) that emerge from the system, each tied to particular eigenvalues.
• In your problem, they take the form \( \sin(nx) \), where \( n \) is an integer.
• The choice of eigenfunctions often depends on an assumed "normalization" (choosing constants for simplicity), such as taking the amplitude \( B = 1 \).

These functions are like musical notes generated by an instrument—distinct, intrinsic modes of vibration. They tell you how a system like a drum skin or a beam can naturally oscillate, which is incredibly useful in engineering and physics.