Please show steps. Use an energy argument to show that the eigenvalues of the Sturm-Liouville Problem...
#2 ONLY PLEASE 1. Consider the non-Sturm-Liouville differential equation Multiply this equation by H(x). Determine H(x) such that the equation may be reduced to the standard Sturm-Liouville form: do Given a(z), 3(2), and 7(2), what are p(x), σ(x), and q(x) 2. Consider the eigenvalue problem (a) Use the result from the previous problem to put this in Sturm-Liouville form (b) Using the Rayleigh quotient, show that λ > 0. (c) Solve this equation subject to the boundary conditions and determine...
II. 1. Find the eigenvalues and the eigenfunctions for the following Sturm-Liouville problem X"+AX=0, x(0)=0, X'(TT) = 0
(20 points) For the following problem use separation of variables to identify the Sturm-Liouville Problem and its eigenvalues and eigenfunctions. DO NOT solve it. This is a steady state temper- ature problem in a cylinder where the temperature depends on ρ and z only. 4. Up(1,2) = 0, 0<z<1 a(p, 0) 0, u(p, 1) 5, 0 < ρ < 1 (20 points) For the following problem use separation of variables to identify the Sturm-Liouville Problem and its eigenvalues and eigenfunctions....
2) For the Sturm-Liouville eigenvalue problem + λφ-0, dt2 do 0, dc (a) 0 verify the following properties: a) The nth eigenfunction has (n-1) zeros on the open interval 0<x<a b) There are an infinite number of eigenvalues with a smallest, but no largest. c) What does the Rayleigh Quotient say about negative and zero eigenfunctions.
please do (iv) and explain all the steps (4) Though I proved in class the orthogonality of eigenfunctions of the Sturm-Liouville BVP with respect to the weight function o when the Sturm-Liouville operator is regular, the orthogonality condition for eigenfunctions is true for many singular Sturm-Liouville BVP's. In this problem you will see an example. Consider then the singular Sturm-Liouville problem [(1 -u-u -1< r< 1, where u is required to be finite at ±1, meaning that limg+1 u(z) is...
1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint: Use an energy argument-multiply the ODE by y and integrate from r 0 to r R; use integration by parts and use the boundedness at r0 to get the boundary term to vanish. (4.75) which is Bessel's equation. Condition (4.72) leads to the boundary condition y(R)0, (4.76) and we impose the boundedness requirement y(0) bounded (4.77) 1. Show that the eigenvalue problem (4.75-4.77) has no negative eigenvalues. Hint:...
This is the question: 42 CHAPTER 2. BASICS Example 2.15 We consider the one-dimensional Sturm-Liouville eigenvalue problem (2.24) - u"(x) = \u()0<<<, (0) = u(T) = 0, that models the vibration of a homogeneous string of length that is fired at both ends. The eigenvalues and eigenvectors or eigenfunctions of (2.24) are x = k?, ux() = sin ka, KEN Let u" denote the approximation of an (eigen)function u at the grid point Ii, uiuti), Di=ih, 0<i<n +1, h =...
Please show working! If unsure of the answer please leave for someone else. Consider the Sturm-Louiville problem d2 y +2y 0 dr2 (0)0, y(3) 0. With n defined as taking values n 1, 2,3, ..., complete the following. (a) Enter the eigenvalues. An = (b) Enter the eigenfunctions Yn Consider the Sturm-Louiville problem d2 y +2y 0 dr2 (0)0, y(3) 0. With n defined as taking values n 1, 2,3, ..., complete the following. (a) Enter the eigenvalues. An =...
Please show complete and neat steps for all the problems 8. The eigenvalues and corresponding eigenvectors for this matrix are given below. 1 -3 1 b+3c a) Verify that these are indeed the correct and valid eigenvector/eigenvalue combinations for this matrix. x(t) y(t) z(t) Give the complete solution to the differential equation X'- AX, where X b) Please give your answers for x(t), y(t), and z(t) explicitly. solvé if you dont 8. The eigenvalues and corresponding eigenvectors for this matrix...
The answer is given. Please show more detailed steps, thank you. 3. Consider the eigenvalue problem 1<x<2 dx2 y(1)=0,y(2) = 0. dx iwrite it in the standard Sturm-Liouville form. ii) Show that 0 by the Rayleigh Quotient. dx p(x)-x, q(x) = 0, σ(x)-1 According the Raileigh Quotient Any eigenvalue is related to its eigenfunction φ(x) by - x p(x) dr Since the B.C. are ф(1)-0 and ф(2-0, so dx 3. Consider the eigenvalue problem 1