Question

State whether the given boundary value problem is homogeneous or non homogeneous. $$ \left[\left(1+x^{2}\right) y^{\prime}\right]+4 y=0, \quad y(0)=0, \quad y(1)=1 $$

Short answer

Question: Determine whether the following boundary value problem is homogeneous or non-homogeneous: $$ \left[\left(1+x^{2}\right) y^{\prime}\right]+4 y=0 $$ Answer: The given boundary value problem is homogeneous, because the right-hand side of the differential equation is 0.

Step-by-step solution

Identify the differential equation

First, let's identify the given differential equation: $$ \left[\left(1+x^{2}\right) y^{\prime}\right]+4 y=0 $$

Analyze the right-hand side

Next, let's analyze the right-hand side of the given differential equation. In this case, the right-hand side is simply \(0\).

Homogeneous vs. non-homogeneous

Since the right-hand side of the given differential equation is \(0\), we can conclude that the given boundary value problem is homogeneous.

Key concepts

Differential Equation

A differential equation is an equation that involves a function and its derivatives. These equations are fundamental in describing various phenomena like physics, engineering, and even biology. In the given problem, we have\[\left(1+x^{2}\right) y^{\prime} + 4y = 0\]Here, \(y\) represents the function depending on \(x\), and \(y^{\prime}\) is the derivative of \(y\) with respect to \(x\). The term \(1 + x^2\) multiplies the derivative, illustrating how the rate of change of \(y\) depends on \(x\).
When working with differential equations, the main goal is usually to solve for the function \(y\), meaning finding the function that satisfies this equation. Solutions can provide insights into how physical systems behave or how populations grow over time, for example.

Boundary Conditions

Boundary conditions are additional constraints provided alongside differential equations. They specify the values of the solution or its derivatives at particular points. In boundary value problems, these conditions are crucial since they define the solutions that are physically or practically possible. The given problem specifies:

• \(y(0) = 0\)
• \(y(1) = 1\)

These conditions tell us that the function \(y(x)\) must satisfy \(y = 0\) when \(x = 0\) and \(y = 1\) when \(x = 1\).
This kind of setup is common in problems dealing with things like temperature distribution along a rod at certain points, or deflection of a beam at specific locations. By setting these conditions, we're essentially tethering the function to these points, needing any solution to "pass through" these boundaries.

Homogeneous vs Non-Homogeneous

The concepts of homogeneous and non-homogeneous refer to whether a differential equation or boundary value problem includes a non-zero term independent of the function and its derivatives. In the provided equation:\[\left(1+x^{2}\right) y^{\prime} + 4y = 0\]
The right-hand side is zero. This means no external forces, inputs, or sources are influencing the system, making it a homogeneous equation. Homogeneous equations usually signify natural systems without external interference.
By contrast, a non-homogeneous equation involves non-zero terms that can represent external inputs or forces, such as additional heat applied to a rod. In practical scenarios, real-world problems often have non-homogeneous parts due to outside influences.