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Chapter 11: Problem 2

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

Expert verified
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

01

Identify the differential equation

First, let's identify the given differential equation: $$ \left[\left(1+x^{2}\right) y^{\prime}\right]+4 y=0 $$
02

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 \(0\).
03

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Consider the problem $$ y^{\prime \prime}+\lambda y=0, \quad \alpha y(0)+y^{\prime}(0)=0, \quad y(1)=0 $$ $$ \begin{array}{l}{\text { where } \alpha \text { is a given constant. }} \\\ {\text { (a) Show that for all values of } \alpha \text { there is an infinite sequence of positive eigenvalues. }} \\ {\text { (b) If } \alpha<1, \text { show that all (real) eigenvalues are positive. Show the smallest eigenvalue }} \\\ {\text { approaches zero as } \alpha \text { approaches } 1 \text { from below. }} \\ {\text { (c) Show that } \lambda=0 \text { is an eigenvalue only if } \alpha=1} \\ {\text { (d) If } \alpha>1 \text { , show that there is exactly one negative eigenvalue and that this eigenvalue }} \\ {\text { decreases as } \alpha \text { increases. }}\end{array} $$

Solve the given problem by means of an eigenfunction expansion. $$ y^{\prime \prime}+2 y=-x, \quad y(0)=0, \quad y(1)=0 $$

Consider the problem $$ -\left(x y^{\prime}\right)^{\prime}+\left(k^{2} / x\right) y=\lambda x y $$ $$ y, y^{\prime} \text { bounded as } x \rightarrow 0, \quad y(1)=0 $$ where \(k\) is a positive integer. (a) Using the substitution \(t=\sqrt{\lambda} x,\) show that the given differential equation reduces to Bessel's equation of order \(k\) (see Problem 9 of Section 5.8 ). One solution is \(J_{k}(t) ;\) a second linearly independent solution, denoted by \(Y_{k}(t),\) is unbounded as \(t \rightarrow 0\). (b) Show formally that the eigenvalues \(\lambda_{1}, \lambda_{2}, \ldots\) of the given problem are the squares of the positive zeros of \(J_{k}(\sqrt{\lambda}),\) and that the corresponding eigenfunctions are \(\phi_{n}(x)=\) \(J_{k}(\sqrt{\lambda_{n}} x) .\) It is possible to show that there is an infinite sequence of such zeros. (c) Show that the eigenfunctions \(\phi_{n}(x)\) satisfy the orthogonality relation $$ \int_{0}^{1} x \phi_{m}(x) \phi_{n}(x) d x=0, \quad m \neq n $$ (d) Determine the coefficients in the formal series expansion $$ f(x)=\sum_{n=1}^{\infty} a_{n} \phi_{n}(x) $$ (e) Find a formal solution of the nonhomogeneous problem $$ -(x y)^{\prime}+\left(k^{2} / x\right) y=\mu x y+f(x) $$ $$ y, y^{\prime} \text { bounded as } x \rightarrow 0, \quad y(1)=0 $$ where \(f\) is a given continuous function on \(0 \leq x \leq 1,\) and \(\mu\) is not an cigenvalue of the corresponding homogeneous problem.

In this problem we outline a proof of the first part of Theorem 11.2 .3 : that the eigenvalues of the Sturm-Liouville problem ( 1 ), (2) are simple. For a given \(\lambda\) suppose that \(\phi_{1}\) and \(\phi_{2}\) are two linearly independent eigenfunctions. Compute the Wronskian \(W\left(\phi_{1}, \phi_{2}\right)(x)\) and use the boundary conditions ( 2) to show that \(W\left(\phi_{1}, \phi_{2}\right)(0)=0 .\) Then use Theorems 3.3 .2 and 3.3 .3 to conclude that \(\phi_{1}\) and \(\phi_{2}\) cannot be linearly independent as assumed.

deal with column buckling problems. In some buckling problems the eigenvalue parameter appears in the boundary conditions as well as in the differential equation. One such case occurs when one end of the column is clamped and the other end is free. In this case the differential equation \(y^{i v}+\lambda y^{\prime \prime}=0\) must be solved subject to the boundary conditions $$ y(0)=0, \quad y^{\prime}(0)=0, \quad y^{\prime \prime}(L)=0, \quad y^{\prime \prime \prime}(L)+\lambda y^{\prime}(L)=0 $$ Find the smallest eigenvalue and the corresponding eigenfunction.
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