Chapter 11: Problem 3
State whether the given boundary value problem is homogeneous or non homogeneous. $$ y^{\prime \prime}+4 y=\sin x, \quad y(0)=0, \quad y(1)=0 $$
Short Answer
Expert verified
Answer: The boundary value problem is non-homogeneous.
Step by step solution
Over 22 million students worldwide already upgrade their learning with Vaia!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Differential Equations
Differential equations are a mathematical way to describe the relationship between a function and its derivatives. Think of them like a recipe that tells you how something changes over time or space. They can describe a wide range of phenomena, including how populations grow, how heat distributes in a room, or how a spring oscillates.
In simple terms, a differential equation will contain an unknown function, let's say \( y \), that we are trying to find, and its derivatives, represented as \( y' \) for the first derivative, \( y'' \) for the second derivative, and so on. The equation will equal some function of \( x \), or it could be just zero. When we solve a differential equation, we are looking for the mystery function \( y \) that makes the equation true for all values of \( x \).
To understand the differential equation presented in the exercise, \( y'' + 4y = \(sin x\) \), imagine a scenario where you're studying the movement of a pendulum. This equation could be telling you how the position of the pendulum changes over time.
In simple terms, a differential equation will contain an unknown function, let's say \( y \), that we are trying to find, and its derivatives, represented as \( y' \) for the first derivative, \( y'' \) for the second derivative, and so on. The equation will equal some function of \( x \), or it could be just zero. When we solve a differential equation, we are looking for the mystery function \( y \) that makes the equation true for all values of \( x \).
To understand the differential equation presented in the exercise, \( y'' + 4y = \(sin x\) \), imagine a scenario where you're studying the movement of a pendulum. This equation could be telling you how the position of the pendulum changes over time.
Boundary Conditions
Now, let's discuss boundary conditions. They are additional pieces of information that specify the value of the solution of a differential equation at certain points. Think of them as rules that the solution must follow at the start and/or end points, much like rules in a game. For example, if we're still imagining our pendulum, a boundary condition might tell us where the pendulum is when we first start the clock.
In the context of our exercise, the boundary conditions are \( y(0) = 0 \) and \( y(1) = 0 \). This means that at the start (when \( x = 0 \)), the solution \( y \) must be zero, and it must again be zero at the end (when \( x = 1 \)). Boundary conditions are crucial when solving differential equations because they often allow us to pin down the exact solution we're after, eliminating other possible functions that don't meet these 'rules'.
In the context of our exercise, the boundary conditions are \( y(0) = 0 \) and \( y(1) = 0 \). This means that at the start (when \( x = 0 \)), the solution \( y \) must be zero, and it must again be zero at the end (when \( x = 1 \)). Boundary conditions are crucial when solving differential equations because they often allow us to pin down the exact solution we're after, eliminating other possible functions that don't meet these 'rules'.
Homogeneous vs Non-Homogeneous
In the world of differential equations, we categorize problems as either homogeneous or non-homogeneous. Homogeneous problems are simpler and come with a uniformity that lets us know that if we multiply a solution by any constant, we will still have a solution. They are characterized by having zero on one side of the equation after setting the function and its derivatives to zero.
On the other hand, non-homogeneous problems, like the one in our exercise, are a bit more complex. They have something other than zero on the opposite side of the differential equation, like the \( \sin x \) in \( y'' + 4y = \sin x \). This little twist means that the solutions to these problems aren't as straightforward, as we need to deal with an extra piece, the non-zero term, which represents some external influence acting on the system.
A quick way to spot a non-homogeneous equation is to look for any terms that don't involve the function being solved for or its derivatives. If you find any function of the independent variable—like \( \sin x \) in our case—that's your clue that the problem is non-homogeneous.
On the other hand, non-homogeneous problems, like the one in our exercise, are a bit more complex. They have something other than zero on the opposite side of the differential equation, like the \( \sin x \) in \( y'' + 4y = \sin x \). This little twist means that the solutions to these problems aren't as straightforward, as we need to deal with an extra piece, the non-zero term, which represents some external influence acting on the system.
A quick way to spot a non-homogeneous equation is to look for any terms that don't involve the function being solved for or its derivatives. If you find any function of the independent variable—like \( \sin x \) in our case—that's your clue that the problem is non-homogeneous.
