Problem 36 - Advanced Engineering Mathematics

Question

Show that the eigenvalues and eigenfunctions of the boundaryvalue problem $$ y^{\prime \prime}+\lambda y=0, \quad y(0)=0, \quad y(1)+y^{\prime}(1)=0 $$ are \(\lambda_{n}=\alpha_{n}^{2}\) and \(y_{n}=\sin \alpha_{n} x\), respectively, where \(\alpha_{n}, n=1\), \(2,3, \ldots\) are the consecutive positive roots of the equation \(\tan \alpha=-\alpha\)

Step-by-Step Solution

Answer

Eigenvalues are \( \lambda_n = \alpha_n^2 \) and eigenfunctions are \( y_n = \sin(\alpha_n x) \) with roots \( \alpha_n \) from \( \tan\alpha = -\alpha \).

Step 1: Formulate the ODE Solution

Consider the differential equation \( y'' + \lambda y = 0 \). Assuming a solution of the form \( y(x) = A \, \cos (\alpha x) + B \, \sin (\alpha x) \), where \( \alpha = \sqrt{\lambda} \). This is a standard solution for linear second-order differential equations with constant coefficients.

Step 2: Apply the Boundary Condition at x=0

Apply the boundary condition \( y(0) = 0 \). Substitute \( x = 0 \) into the general solution: \( A \, \cos(0) + B \, \sin(0) = 0 \Rightarrow A = 0 \). Therefore, the solution simplifies to \( y(x) = B \, \sin(\alpha x) \).

Step 3: Apply the Boundary Condition at x=1

Now apply the second boundary condition \( y(1) + y'(1) = 0 \). Compute \( y'(x) = B \, \alpha \, \cos(\alpha x) \). At \( x = 1 \), \( y(1) = B \, \sin(\alpha) \) and \( y'(1) = B \, \alpha \, \cos(\alpha) \). Thus, \[ B \, \sin(\alpha) + B \, \alpha \, \cos(\alpha) = 0. \] Factor out \( B \) (assuming \( B eq 0 \)), obtaining \( \sin(\alpha) + \alpha \, \cos(\alpha) = 0 \).

Step 4: Solve the Characteristic Equation

Rewrite the equation \( \sin(\alpha) + \alpha \, \cos(\alpha) = 0 \) as \( \tan(\alpha) = -\alpha \). The roots \( \alpha_n \) of this equation determine the eigenvalues and corresponding eigenfunctions. Thus, the eigenvalues are given by \( \lambda_n = \alpha_n^2 \), with eigenfunctions \( y_n(x) = \sin(\alpha_n x) \).

Step 5: Conclusion: Verify and Summarize Results

Thus, we have shown that the eigenvalues are \( \lambda_n = \alpha_n^2 \) and the eigenfunctions are \( y_n = \sin(\alpha_n x) \), corresponding to the positive roots \( \alpha_n \) of \( \tan(\alpha) = -\alpha \). Each \( \alpha_n \) corresponds to a different eigenvalue and eigenfunction.

Key Concepts

Eigenvalues

In the context of a boundary value problem, eigenvalues are special numbers that allow a differential equation to have non-trivial solutions. Understanding eigenvalues is crucial for solving many physical and engineering problems. When you solve a differential equation, such as \[ y'' + \lambda y = 0, \]with particular boundary conditions given, the values of \( \lambda \) that result in valid solutions are its eigenvalues.Why Eigenvalues Matter:

• They provide key insights into the stability and dynamics of systems.
• In vibration analysis, they can signify natural frequencies.

For the problem at hand, the eigenvalues are determined by the equation\[ \tan(\alpha) = -\alpha, \]where \( \alpha = \sqrt{\lambda} \). The roots of this equation, \( \alpha_n \), determine the eigenvalues as \( \lambda_n = \alpha_n^2 \). This step shows how eigenvalues help to build a bridge between abstract mathematical solutions and real-world applications.

Eigenfunctions

Eigenfunctions are the corresponding functions that arise from solving a boundary value problem. When these functions are plugged into the differential equation along with an eigenvalue, they satisfy both the differential equation and the boundary conditions.Characteristics of Eigenfunctions:

• They are non-zero solutions that accompany each eigenvalue.
• They satisfy all given conditions of the boundary value problem.

For the particular boundary value problem\[ y'' + \lambda y = 0, \quad y(0) = 0, \quad y(1) + y'(1) = 0, \]the eigenfunctions are of the form\[ y_n(x) = \sin(\alpha_n x), \]where \( \alpha_n \) are the roots derived from \( \tan(\alpha) = -\alpha \). These eigenfunctions demonstrate how patterns or modes form in many physical phenomena, especially in wave and oscillation scenarios.

Differential Equations

Differential equations are equations that involve derivatives of a function. They are powerful tools used to describe a wide variety of physical systems. In boundary value problems, differential equations are accompanied by conditions set at the boundaries of the domain.Types and Importance:

• Ordinary Differential Equations (ODEs): involve functions of a single variable.
• Partial Differential Equations (PDEs): involve functions of multiple variables.

The equation \( y'' + \lambda y = 0 \) is an ordinary differential equation signaling a system whose rate of change depends on its current state.Why They are Essential:They model real-world systems such as:

• Mechanics and Motion
• Thermodynamics and Heat Transfer

In this exercise, the differential equation helps to signify locations where the slope or curvature of the function changes, thereby leading to solutions that must meet specific boundary conditions.

Boundary Conditions

Boundary conditions are restrictions or constraints placed on differential equations to find unique solutions. They specify the values that an unknown function has to satisfy at the boundaries of the domain.Significance of Boundary Conditions:

• They determine the shape and behavior of the solution across the domain.
• They ensure the solution fits the physical scenario modeled by the equation.

In the given problem, the boundary conditions are:\[ y(0) = 0, \quad y(1) + y'(1) = 0. \]These conditions dictate how the solution behaves at \( x = 0 \) and \( x = 1 \). The first condition \( y(0) = 0 \) ensures that the function is zero at the starting point, while the second condition \( y(1) + y'(1) = 0 \) mixes the function’s value and its rate of change at the endpoint. They are fundamentally vital to find the specific form of eigenfunctions, leading to meaningful real-world interpretations.