Problem 14 - Advanced Engineering Mathematics

Question

In Problems 11-20, find the eigenvalues and eigenfunctions for the given boundary-value problem. $$ y^{\prime \prime}+\lambda y=0, y(0)=0, \quad y^{\prime}(\pi / 2)=0 $$

Step-by-Step Solution

Answer

Eigenvalues are \( \lambda_n = (2n+1)^2 \); eigenfunctions are \( y_n(x) = \sin((2n+1)x) \).

Step 1: Understand the Problem

We need to find the eigenvalues \( \lambda \) and associated eigenfunctions \( y(x) \) for the differential equation \( y'' + \lambda y = 0 \) subject to the boundary conditions \( y(0) = 0 \) and \( y'(\pi/2) = 0 \).

Step 2: Solve the Differential Equation

The differential equation \( y'' + \lambda y = 0 \) has solutions depending on the sign of \( \lambda \). For now, assume \( \lambda = k^2 \) where \( k \) can be real or imaginary. We have:- If \( \lambda = k^2 > 0 \), the solution is \( y(x) = A \cos(kx) + B \sin(kx) \).- If \( \lambda = -k^2 < 0 \), the solution is \( y(x) = A e^{kx} + B e^{-kx} \).- If \( \lambda = 0 \), the solution is \( y(x) = Ax + B \).

Step 3: Apply Boundary Condition at x=0

Apply the first boundary condition \( y(0) = 0 \). For the trigonometric case \( y(x) = A \cos(kx) + B \sin(kx) \):\( y(0) = A \cos(0) + B \sin(0) = A = 0 \).Thus, \( A = 0 \), leaving \( y(x) = B \sin(kx) \).

Step 4: Apply Boundary Condition at x=π/2

Apply the second boundary condition \( y'(\pi/2) = 0 \). Compute the derivative:\( y'(x) = Bk \cos(kx) \).Thus, \( y'(\pi/2) = Bk \cos(k\pi/2) = 0 \).For \( Bk eq 0 \), \( \cos(k\pi/2) = 0 \) implies \( k\pi/2 = (2n+1)\pi/2 \) for odd \( n \). Thus, \( k = 2n+1 \).

Step 5: Determine Eigenvalues

Since \( k = 2n + 1 \), the eigenvalues are \( \lambda_n = (2n + 1)^2 \) where \( n \) is a non-negative integer.

Step 6: Determine Eigenfunctions

The corresponding eigenfunctions for each eigenvalue \( \lambda_n = (2n + 1)^2 \) are \( y_n(x) = B \sin((2n + 1)x) \), where \( B \) is a constant, usually set for normalization.

Key Concepts

Boundary-Value Problems

Boundary-value problems are an important class of problems in differential equations. These involve finding a solution to a differential equation that satisfies specific conditions at the endpoints of the interval. In our exercise, we are looking into a boundary-value problem where the function has certain required values at the boundaries of the interval under consideration (for example, at \( x = 0 \) and \( x = \pi/2 \)). This type of problem is particularly prevalent in physics and engineering because they often correspond to real-world scenarios such as heat distribution or vibration modes.

• The boundary condition \( y(0) = 0 \) specifies the function's value at \( x = 0 \).
• The condition \( y'(\pi/2) = 0 \) specifies the slope of the function at \( x = \pi/2 \).

These conditions allow us to determine specific solutions from a family of possible solutions, guiding us to the so-called eigenvalues and eigenfunctions.

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They are a tool used to describe and understand how quantities change over time or space, which makes them ubiquitous in science and engineering. In this exercise, we are dealing with a second-order linear homogeneous differential equation of the form \[ y'' + \lambda y = 0 \]. The potential solutions depend on the parameter \( \lambda \), which we call an eigenvalue:

• When \( \lambda = k^2 > 0 \), solutions take the form of trigonometric functions like \( \sin \) and \( \cos \).
• For \( \lambda = -k^2 < 0 \), solutions resemble exponential functions.
• In the case where \( \lambda = 0 \), the solution is linear.

By solving this equation with prescribed boundary conditions, we determine not only the values of \( \lambda \) that allow non-trivial solutions but also establish the form of these solutions, resulting in the eigenfunctions we need.

Eigenfunctions

Eigenfunctions are specific solutions to a differential equation that correspond to particular eigenvalues. In the context of boundary-value problems, these solutions satisfy both the differential equation and the boundary conditions. The term "eigenfunction" comes from the German word "eigen," meaning "own" or "characteristic," highlighting that these functions are intrinsically linked to the system described by the differential equation.

• In the provided solution, \( y_n(x) = B \sin((2n + 1)x) \) are the eigenfunctions derived after applying boundary conditions.
• Each eigenfunction corresponds to a specific eigenvalue \( \lambda_n = (2n + 1)^2 \).
• The constant \( B \) is often set to normalize the eigenfunctions, which can be particularly useful in applications involving physical systems.

Understanding eigenfunctions helps us predict the behavior of complex systems, such as vibrations or waves, identifying the characteristics that are fundamental to that system's behavior.