2) For the Sturm-Liouville eigenvalue problem + λφ-0, dt2 do 0, dc (a) 0 verify the...
#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...
6. Consider the eigenvalue problem 1 < x < 2, y(1) = 0, y(2) = 0. (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain |(c) Is the operator S symmetric? Explain. (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 (e) Find the orthogonal expansion of f(x) = ln x, 1 < x < 2, in terms of these eigenfunctions. (f) Find the smallest N such that...
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
II. 1. Find the eigenvalues and the eigenfunctions for the following Sturm-Liouville problem X"+AX=0, x(0)=0, X'(TT) = 0
5. Consider the problem a2y"y _2.J 0 x1 = 0, y(0) 0, y(1= 0. (a) Put the problem in Sturm-Liouville form and explain the nature of any singular points. (b) State the appropriate modified boundary conditions (c) Find all eigenvalues and eigenfunctions for the modified problem 5. Consider the problem a2y"y _2.J 0 x1 = 0, y(0) 0, y(1= 0. (a) Put the problem in Sturm-Liouville form and explain the nature of any singular points. (b) State the appropriate modified...
(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....
just put the eigenvalue and eigenfunctions would be good Consider the Sturm-Louiville problem d2y Ay 0, y0) 0, (5) 0. dr2 With n defined as taking values n = 1,2, 3, .., complete the following (a) Enter the eigenvalues. (b) Enter the eigenfunctions. yn = Consider the Sturm-Louiville problem d2y Ay 0, y0) 0, (5) 0. dr2 With n defined as taking values n = 1,2, 3, .., complete the following (a) Enter the eigenvalues. (b) Enter the eigenfunctions. yn...
Solve part (d) 6. Consider the eigenvalue problem 2"xy3y Ay 0 y(1)0, y(2)= 0. + 1 < x< 2, (a) Write the problem in Sturm-Liouville form, identifying p, q, and w. (b) Is the problem regular? Explain (c) Is the operator S symmetric? Explain (d) Find all eigenvalues and eigenfunctions. Discuss in light of Theorem 4.3 ln x, 1 < 2, in terms of these (e) Find the orthogonal expansion of f(x) eigenfunctions _ 6. Consider the eigenvalue problem 2"xy3y...
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 solve all 3 Differential Equation problems 3.8.7 Question Help Consider the following eigenvalue problem for which all of its eigenvalues are nonnegative y',thy-0; y(0)-0, y(1) + y'(1)-0 (a) Show that λ =0 is not an eigenvalue (b) Show that the eigenfunctions are the functions {sin α11,o, where αη įs the nth positive root of the equation tan z -z (c) Draw a sketch indicating the roots as the points of intersection of the curves y tan z and y...