Problem 15 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

Eigenvalues are \( \lambda_n = \left(n + \frac{1}{2} \right)^2 \); eigenfunctions are \( y_n(x) = A \cos((n + \frac{1}{2})x) \).

Step 1: Define the Differential Equation

We are given the differential equation: \[ y'' + \lambda y = 0 \] where \( \lambda \) is a parameter that will be determined such that there are nontrivial solutions to the differential equation satisfying the given boundary conditions.

Step 2: Analyze the Boundary Conditions

The given boundary conditions are: \[ y'(0) = 0 \quad \text{and} \quad y(\pi) = 0 \]These conditions will help us determine the eigenvalues \( \lambda \) and the corresponding eigenfunctions.

Step 3: Solve the Characteristic Equation

Assume a solution of the form \( y(x) = A \cos(\sqrt{\lambda} x) + B \sin(\sqrt{\lambda} x) \). Find the derivative: \[ y'(x) = -A \sqrt{\lambda} \sin(\sqrt{\lambda} x) + B \sqrt{\lambda} \cos(\sqrt{\lambda} x) \] Apply the condition \( y'(0) = 0 \) to get: \[ B \sqrt{\lambda} = 0 \] Thus, \( B = 0 \), implying \( y(x) = A \cos(\sqrt{\lambda} x) \).

Step 4: Apply the Second Boundary Condition

Using the second condition \( y(\pi) = 0 \), substitute into the remaining solution: \[ A \cos(\sqrt{\lambda} \pi) = 0 \] This implies \( \cos(\sqrt{\lambda} \pi) = 0 \), leading to \( \sqrt{\lambda} \pi = (2n+1)\frac{\pi}{2} \) for integer \( n \).

Step 5: Calculate the Eigenvalues

From \( \sqrt{\lambda} \pi = (2n+1)\frac{\pi}{2} \), solve for \( \lambda \): \[ \lambda = \left(\frac{(2n+1)\pi}{2\pi} \right)^2 = \left(\frac{2n+1}{2} \right)^2 \] Thus, the eigenvalues are \[ \lambda_n = \left(n + \frac{1}{2} \right)^2 \] for \( n = 0, 1, 2, \ldots \)

Step 6: Determine the Eigenfunctions

With \( \lambda_n = \left(n + \frac{1}{2} \right)^2 \), the eigenfunctions corresponding to each eigenvalue \( \lambda_n \) are: \[ y_n(x) = A \cos\left(\left(n + \frac{1}{2} \right)x\right) \] where \( A \) is a nonzero constant.

Key Concepts

Differential Equations

Differential equations are mathematical equations that relate a function with its derivatives. They are fundamental in understanding how quantities change over continuous domains. The core of our exercise involves the differential equation: \[ y'' + \lambda y = 0 \] In this context, \( y \) is a function of \( x \), and \( y'' \) is its second derivative. The parameter \( \lambda \) is an important part of the equation and is determined by the boundary conditions. Solving differential equations involves finding functions that satisfy both the equation and the additional conditions provided. These solutions have a wide range of applications in physics, engineering, and other sciences. They help model systems where change is continuous, such as the vibration of strings or the flow of heat.

Boundary Conditions

Boundary conditions are essential to the solution of differential equations. They specify the values that a solution must satisfy at the boundaries of the domain. In our problem, the boundary conditions are:

• \( y'(0) = 0 \)
• \( y(\pi) = 0 \)

These conditions restrict the form of possible solutions for the differential equation. Applying these conditions allows us to find the specific values of \( \lambda \) that lead to non-trivial solutions.
These values of \( \lambda \) are called eigenvalues. Including boundary conditions means we seek solutions that fit not just the differential equation, but also specific criteria elsewhere, often at the start and end of an interval. They are particularly important in physical problems where constraints naturally exist at the boundaries.

Eigenfunctions

Eigenfunctions are solutions to differential equations that are related to specific eigenvalues. In our context, after applying the boundary conditions, we derive that the eigenfunctions take the form:

• \[ y_n(x) = A \cos\left(\left(n + \frac{1}{2} \right)x\right) \]

Here, \( A \) is a constant and \( n \) is an integer, giving rise to different solutions for different \( n \). Each \( y_n(x) \) is associated with an eigenvalue \( \lambda_n = \left(n + \frac{1}{2} \right)^2 \). This relationship is fundamental in problems of wave mechanics and quantum systems, where eigenfunctions represent modes of the system. They highlight the shapes or patterns that certain parameters will take under specified conditions. These functions are indispensable in simplifying complex systems into understandable patterns.