Question

In each of Problems I through 6 state whether the given boundary value problem is homogeneous or non homogeneous. $$ y^{\prime \prime}+4 y=0, \quad y(-1)=0, \quad y(1)=0 $$

Short answer

Differential equation: \(y'' + 4y = 0\) Boundary conditions: \(y(-1) = 0\) and \(y(1) = 0\) Answer: The given boundary value problem is homogeneous.

Step-by-step solution

Identify the Differential Equation and Boundary Conditions

The given boundary value problem can be represented as: Differential equation: \(y'' + 4y = 0\) Boundary conditions: \(y(-1) = 0\) and \(y(1) = 0\)

Check if the Differential Equation is Homogeneous

The differential equation is given by \(y'' + 4y = 0\). Since it is already in the required form Ly = 0, we can conclude that the differential equation is homogeneous.

Check if the Boundary Conditions are Homogeneous

The given boundary conditions are \(y(-1) = 0\) and \(y(1) = 0\). Since both conditions are in the form \(y(x) = 0\), we can conclude that the boundary conditions are homogeneous.

Determine if the Boundary Value Problem is Homogeneous or Nonhomogeneous

Since the differential equation and boundary conditions are both homogeneous, the given boundary value problem is a homogeneous one.

Key concepts

Homogeneous Differential Equation

In mathematics, a homogeneous differential equation is one where all terms are either zero or they can be expressed in terms of multiples of the unknown function and its derivatives.
This means that the right hand side of the equation is zero. For example, in the equation \(y'' + 4y = 0\), each term involves the function \(y\) or its derivatives. No external force or non-zero terms are added to the equation.
Homogeneous differential equations are important because they often describe natural phenomena like vibrating systems or electrical circuits without external influences.

Differential Equation

A differential equation involves the relationship between a function and its derivatives. These equations are essential in expressing physical principles and phenomena where a rate of change is involved.
For instance, the equation \(y'' + 4y = 0\) involves a second-order derivative, \(y''\), which indicates a relationship between a function \(y\) and its rate of change.
Differential equations play a crucial role in a wide range of fields including physics, engineering, and economics, offering a powerful tool to model dynamic systems.

Boundary Conditions

Boundary conditions are additional constraints necessary to solve a differential equation uniquely. They specify the value of the solution at certain points, providing the needed information to narrow down possible solutions.
In the given boundary value problem, \(y(-1) = 0\) and \(y(1) = 0\) are the boundary conditions. These conditions stipulate that the solution function \(y(x)\) must equal zero at \(x = -1\) and \(x = 1\).
Applying boundary conditions is essential in ensuring the solution's relevance to the real-world scenario being modeled.

Mathematical Analysis

Mathematical analysis involves analyzing equations, their behaviors, and solutions using rigorous mathematics. It provides tools for understanding the properties of solutions to equations, especially differential equations.
Through analysis, we can determine whether a differential equation is homogeneous. By studying the boundary conditions, we ascertain that the boundary value problem reflects a real-world problem accurately.
Mathematical analysis ensures that we comprehend the full scope of a problem, enabling effective prediction and solution development through careful examination of equations' behaviors and solutions.