Question

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

Short answer

Differential equation: \(- y^{\prime \prime} + x^{2}y = \lambda y\) Boundary conditions: \(y^{\prime}(0) - y(0)=0\) and \(y^{\prime}(1) + y(1)=0\) Answer: The given boundary value problem is homogeneous.

Step-by-step solution

Analyze the differential equation

The given differential equation is: $$ - y^{\prime \prime} + x^{2}y = \lambda y $$ This is a linear differential equation. To determine if it's homogeneous or not, we need to examine the right-hand side of the equation, which is \(\lambda y\). Since \(\lambda y\) does not involve any function other than \(y\), this equation is homogeneous.

Analyze the boundary conditions

Now let's examine the boundary conditions: 1. \(y^{\prime}(0) - y(0)=0\) 2. \(y^{\prime}(1) + y(1)=0\) For boundary condition 1: $$ y^{\prime}(0) - y(0) = 0 \implies y^{\prime}(0) = y(0) $$ Since this equation is true when \(y^{\prime}(0)=0\) and \(y(0)=0\), the first boundary condition is homogeneous. For boundary condition 2: $$ y^{\prime}(1) + y(1) = 0 \implies y^{\prime}(1) = -y(1) $$ Similar to the first boundary condition, this equation is true when \(y^{\prime}(1)=0\) and \(y(1)=0\). Therefore, the second boundary condition is also homogeneous.

Determine if the boundary value problem is homogeneous or non-homogeneous

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

Key concepts

Homogeneous Differential Equation

A homogeneous differential equation is one where, generally, each term is a function of the dependent variable and its derivatives. This means every term involves the function or its derivatives in a linear manner without any external additive terms.

In the exercise provided, the differential equation is \(- y'' + x^2 y = \lambda y\). We look at the right-hand side, \(\lambda y\), which indeed involves only the function \(y\) multiplied by \(\lambda\), a parameter or constant.

This indicates that there are no terms involving other functions or non-constant terms that do not include \(y\). Thus, the equation is homogeneous.

Essential characteristics of homogeneous differential equations include:

• No constant or specific external forces act on the system within the equation.
• Solutions can often be expressed as combinations of functions, known as superpositions.
• The zero function (\(y = 0\) everywhere) is always a solution.

Linear Differential Equation

Linear differential equations are a type of equation where the dependent variable \(y\) and all its derivatives appear to the first degree and are not multiplied by one another. This makes them "linear" in nature, similar to linear algebra where variables are raised only to the first power.

In our exercise, the linearity is evident as \(- y'' + x^2 y = \lambda y\); every term involving \(y\) and its derivative appears linearly.

These equations are very useful due to:

• Simplicity - They are easier to solve compared to non-linear equations.
• Superposition - The principle of superposition applies, allowing solutions to be added together to form new solutions.
• Predictability - Their behavior is well-understood and predictable.

The trait of being linear is crucial for deciding solution methods and understanding the system being described.

Boundary Conditions

Boundary conditions are constraints necessary for solving differential equations that depict real-world situations. They define how the solution behaves at the edges of the domain and ensure that the solution suits the physical boundaries of the problem.

In the provided exercise, we have two boundary conditions:
1. \(y'(0) - y(0) = 0\).
2. \(y'(1) + y(1) = 0\).

For these conditions to be homogeneous, they need to hold when \(y\) is zero over its entire domain.

These conditions assist in:

• Ensuring uniqueness of solutions, meaning the same boundary value problem does not have multiple solutions.
• Reflecting the physical constraints or symmetries of the system being studied.
• Allowing for the identification of only solutions that are physically feasible or meaningful in context.

Essentially, boundary conditions are crucial for fully understanding and resolving differential equations in applicable scenarios.