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Chapter 11: Problem 6

State whether the given boundary value problem is homogeneous or non homogeneous. $$ -y^{\prime \prime}=\lambda\left(1+x^{2}\right) y, \quad y(0)=0, \quad y^{\prime}(1)+3 y(1)=0 $$

Short Answer

Expert verified
Answer: The given Boundary Value Problem is homogeneous, as both the boundary conditions are homogeneous.

Step by step solution

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01

Analyze the first boundary condition

The first boundary condition is given by: \(y(0) = 0\). This condition tells us that the dependent variable y is equal to 0 at x=0, and since there are no other functions or constants involved in this condition, it is homogeneous.
02

Analyze the second boundary condition

The second boundary condition is given by: \(y^{\prime}(1) + 3y(1) = 0\). In this case, we have both the function's derivative, i.e., \(y^{\prime}(1)\), and the function itself, i.e., \(y(1)\) in the condition. There are no other constants or functions involved in this condition, implying that it is also homogeneous.
03

Determine the overall homogeneity of the BVP

Since both boundary conditions are homogeneous, the given Boundary Value Problem is a homogeneous boundary value problem.

Key Concepts

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

Homogeneous Differential Equations
A homogeneous differential equation can be thought of as an equation where the result is zero when all the terms of the function and its derivatives are incorporated. This indicates there are no additional additions of constants or functions that shift the solution away from zero. Homogeneous equations typically have the general form:
  • \( a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \ldots + a_0(x)y = 0 \)
where \(a_n(x)\), \(a_{n-1}(x)\), etc., are functions of the independent variable \(x\).
In solving these equations, the focus is on the combination of the function and its derivatives, which together must satisfy the equation to equal zero. The absence of distinct non-homogeneous terms means the focus is solely on how these mathematical increments – the function and its derivatives – influence each other to meet that requirement.
Hence, when dealing with boundary value problems designated as homogeneous, such as the one in the original exercise \(-y'' = \lambda(1+x^2)y\), every aspect of the equation fundamentally couples with the conditions set without deviation through additional constants or external terms.
Boundary Conditions
Boundary conditions are crucial in determining the solutions to differential equations within a specific interval. They represent additional information that allows us to capture the behavior of the solution at the boundaries of this interval.
There are two principal types often discussed:
  • Dirichlet Condition: This specifies the values a solution must take at the boundary. For example, \(y(0) = 0\) in our exercise specifies \(y\) is zero when \(x\) is zero.
  • Neumann Condition: Specifies the values of the derivative at the boundary. The condition \(y'(1) + 3y(1) = 0\) combines such a requirement with the value of the function itself at the boundary.
Together, these conditions bound the differential equation and help in characterizing the exact nature of solutions that will satisfy both the equation and these boundary demands consistently across the given domain.
Second Order Differential Equations
A second order differential equation involves the second derivative of the function, indicating how the function changes at a rate that depends on more than just its initial condition. Its general form includes terms up to the second derivative:
  • \( ay'' + by' + cy = f(x) \)
where \( y'' \) denotes the second derivative of \( y \), and \( f(x) \) is some function of \( x \).
Such equations are prevalent in physical systems modeling, where they describe things like motion, heat distributions, or oscillations. In the exercise provided, the equation is:
  • \( -y'' = \lambda(1 + x^2)y \)
This includes the term \(-y''\), capturing the behavior of the function's second derivative directly impacting its form along with the other terms present. Solving such equations typically requires techniques that consider these multivariate dependencies and often the context given by boundary conditions, to ensure the deduced solution reflects both the intrinsic dynamics of the modeled scenario and the constraints applied.

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Most popular questions from this chapter

Consider the Sturm-Liouville problem $$ -\left[p(x) y^{\prime}\right]^{\prime}+q(x) y=\lambda r(x) y $$ $$ a_{1} y(0)+a_{2} y^{\prime}(0)=0, \quad b_{1} y(1)+b_{2} y^{\prime}(1)=0 $$ where \(p, q,\) and \(r\) satisfy the conditions stated in the text. (a) Show that if \(\lambda\) is an eigenvalue and \(\phi\) a corresponding eigenfunction, then $$ \lambda \int_{0}^{1} r \phi^{2} d x=\int_{0}^{1}\left(p \phi^{2}+q \phi^{2}\right) d x+\frac{b_{1}}{b_{2}} p(1) \phi^{2}(1)-\frac{a_{1}}{a_{2}} p(0) \phi^{2}(0) $$ provided that \(a_{2} \neq 0\) and \(b_{2} \neq 0 .\) How must this result be modified if \(a_{2}=0\) or \(b_{2}=0\) ? (b) Show that if \(q(x) \geq 0\) and if \(b_{1} / b_{2}\) and \(-a_{1} / a_{2}\) are nonnegative, then the eigenvalue \(\lambda\) is nonnegative. (c) Under the conditions of part (b) show that the eigenvalue \(\lambda\) is strictly positive unless \(q(x)=0\) for each \(x\) in \(0 \leq x \leq 1\) and also \(a_{1}=b_{1}=0\)

The equation $$ u_{t t}+c u_{t}+k u=a^{2} u_{x x}+F(x, t) $$ where \(a^{2}>0, c \geq 0,\) and \(k \geq 0\) are constants, is known as the telegraph equation. It arises in the study of an elastic string under tension (see Appendix \(\mathrm{B}\) of Chapter 10 ). Equation (i) also occurs in other applications. Assuming that \(F(x, t)=0,\) let \(u(x, t)=X(x) T(t),\) separate the variables in Eq. (i), and derive ordinary differential equations for \(X\) and \(T\)

Differ from those in previous problems in that the parameter \(\lambda\) multiplies the \(y^{\prime}\) term as well as the \(y\) term. In each of these problems determine the real eigenvalues and the corresponding eigenfunctions. $$ \begin{array}{l}{x^{2} y^{\prime \prime}-\lambda\left(x y^{\prime}-y\right)=0} \\\ {y(1)=0, \quad y(2)-y^{\prime}(2)=0}\end{array} $$

Consider the boundary value problem $$ y^{\prime \prime}-2 y^{\prime}+(1+\lambda) y=0, \quad y(0)=0, \quad y(1)=0 $$ $$ \begin{array}{l}{\text { (a) Introduce a new dependent variable } u \text { by the relation } y=s(x) u \text { . Determine } s(x) \text { so }} \\ {\text { that the differential equation for } u \text { has no } u \text { 'term. }} \\\ {\text { (b) Solve the boundary value problem for } u \text { and thereby determine the eigenvalues and }} \\ {\text { eigenfunctions of the original problem. Assume that all eigenvalues are real. }} \\ {\text { (c) Also solve the given problem directly (without introducing } u \text { ). }}\end{array} $$

Consider the boundary value problem $$ -\left(x y^{\prime}\right)^{\prime}=\lambda x y $$ \(y, y^{\prime}\) bounded as \(x \rightarrow 0, \quad y^{\prime}(1)=0\) (a) Show that \(\lambda_{0}=0\) is an eigenvalue of this problem corresponding to the eigenfunction \(\phi_{0}(x)=1 .\) If \(\lambda>0,\) show formally that the eigenfunctions are given by \(\phi_{n}(x)=\) \(J_{0}(\sqrt{\lambda_{n}} x),\) where \(\sqrt{\lambda_{n}}\) is the \(n\) th positive root (in increasing order) of the equation \(J_{0}^{\prime}(\sqrt{\lambda})=0 .\) It is possible to show that there is an infinite sequence of such roots. (b) Show that if \(m, n=0,1,2, \ldots,\) then $$ \int_{0}^{1} x \phi_{m}(x) \phi_{n}(x) d x=0, \quad m \neq n $$ (c) Find a formal solution to the nonhomogeneous problem $$ \begin{aligned}-\left(x y^{\prime}\right)^{\prime} &=\mu x y+f(x) \\ y, y^{\prime} \text { bounded as } x \rightarrow 0, & y^{\prime}(1)=0 \end{aligned} $$ where \(f\) is a given continuous function on \(0 \leq x \leq 1,\) and \(\mu\) is not an eigenvalue of the corresponding homogeneous problem.
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