Problem 14 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

Eigenvalues: \(\lambda = (2n+1)^2\), Eigenfunctions: \(y_n(x) = B \sin((2n+1)x)\).

Step 1: Formulate the Differential Equation

The given boundary-value problem is expressed as the differential equation \(y'' + \lambda y = 0\) with boundary conditions \(y(0) = 0\) and \(y'(\frac{\pi}{2}) = 0\). The goal is to find the eigenvalues (\(\lambda\)) and the corresponding eigenfunctions (\(y(x)\)).

Step 2: Propose a Solution for the Differential Equation

Consider a solution of the form \(y(x) = A \cos(\sqrt{\lambda} x) + B \sin(\sqrt{\lambda} x)\). This is a general solution for a second-order linear homogeneous differential equation with constant coefficients.

Step 3: Apply the Boundary Conditions to Find Constants

First, apply the boundary condition \(y(0) = 0\). Substituting \(x = 0\) into the solution gives \(y(0) = A \cos(0) + B \sin(0) = A = 0\). Thus, \(A = 0\) and the solution simplifes to \(y(x) = B \sin(\sqrt{\lambda} x)\).

Step 4: Use Second Boundary Condition to Determine Eigenvalues

The next boundary condition is \(y'(\frac{\pi}{2}) = 0\). First, compute the derivative: \(y'(x) = B \sqrt{\lambda} \cos(\sqrt{\lambda} x)\). Substituting \(x = \frac{\pi}{2}\), we get \(B \sqrt{\lambda} \cos(\sqrt{\lambda} \frac{\pi}{2}) = 0\). Since \(B eq 0\), it must be that \(\cos(\sqrt{\lambda} \frac{\pi}{2}) = 0\).

Step 5: Solve for Eigenvalues

The equation \(\cos(\frac{\pi}{2}\sqrt{\lambda}) = 0\) implies that \(\frac{\pi}{2}\sqrt{\lambda} = (2n+1)\frac{\pi}{2}\) for \(n = 0, 1, 2, ...\). Solving for \(\lambda\) gives \(\sqrt{\lambda} = 2n+1\), hence \(\lambda = (2n+1)^2\).

Step 6: Define Eigenfunctions for Each Eigenvalue

For each eigenvalue \(\lambda = (2n+1)^2\), the corresponding eigenfunction is \(y_n(x) = B \sin((2n+1)x)\). This satisfies both the differential equation and the boundary conditions.

Key Concepts

Boundary-Value Problem

A boundary-value problem in mathematics refers to a differential equation accompanied by a set of constraints called boundary conditions. These conditions specify values or behaviors of the solution at specific points, such as the edges of an interval. They are crucial for ensuring that there is a unique solution to the problem.
In our given exercise, the boundary conditions are \( y(0) = 0 \) and \( y'(\frac{\pi}{2}) = 0 \). These conditions must be satisfied by the solution of the differential equation. Boundary-value problems frequently arise in physical situations where conditions at the boundaries of a domain influence the behavior of a system. They help mold the potential solutions and restrict them to those that fit the real-world scenarios.
When dealing with boundary-value problems, keep in mind:

• They often relate to spatial problems where solutions are known at the boundaries.
• The existence and uniqueness of solutions depend on both the equation and the boundary conditions.
• They can be found in various fields, including physics, engineering, and finance.

Differential Equation

Differential equations are equations that involve a function and its derivatives. They are essential tools for understanding dynamic systems across various scientific fields. These equations describe how a particular quantity changes with respect to another, capturing the essence of many natural phenomena.
In simple terms, a differential equation links a function with its rate of change. The given problem involves a specific type of differential equation: \( y'' + \lambda y = 0 \). This is a second-order differential equation as it includes a second derivative, \( y'' \). The aim is to find functions \( y(x) \) that satisfy this equation under certain conditions.
Things to remember about differential equations:

• They can be classified by the order of the highest derivative they contain (first-order, second-order, etc.).
• Solving them involves finding the function \( y(x) \) that makes the equation true.
• They are widely used to model real-world problems in science and engineering.

Eigenvalue Problem

An eigenvalue problem is a specific type of mathematical problem important for understanding linear transformations and vibrations. It involves finding scalars (eigenvalues) and associated vectors (eigenfunctions) that satisfy a condition derived from a linear equation.
In the context of our problem, the eigenvalue problem emerges from the differential equation \( y'' + \lambda y = 0 \). Here, \( \lambda \) represents the eigenvalues, which are specific constants that allow non-zero solutions (eigenfunctions) to satisfy the equation and the boundary conditions. In this exercise, solving the eigenvalue problem provides \( \lambda \) as \( (2n+1)^2 \) and corresponding eigenfunctions \( y_n(x) = B \sin((2n+1)x) \).
Key points about eigenvalue problems:

• Eigenvalues \( \lambda \) determine the characteristic behavior of the system described by the equation.
• Eigenfunctions \( y(x) \) are the solutions (functions) that accompany each eigenvalue.
• Such problems are essential for solving physical problems involving vibrational modes, stability analysis, and more.

Second-Order Differential Equations

Second-order differential equations are a type of differential equation where the second derivative of the function is the highest derivative present. These are fundamental in modeling phenomena where acceleration or curvature is considered, such as mechanical vibrations or electrical circuits.
The equation \( y'' + \lambda y = 0 \) is an example of a second-order linear homogeneous differential equation. The term "homogeneous" implies that there are no additional terms other than those involving the function and its derivatives.
When solving second-order differential equations:

• The general solution is typically a combination of sinusoidal functions (sine and cosine) when the coefficients are constant.
• They often require specific techniques and assumptions to solve, particularly considering boundary conditions.
• Understanding second-order equations is key for analyzing waves, oscillations, and forces in physics and engineering.

These equations are foundational in learning how systems evolve over time and space, proving their worth in a vast array of applied contexts.