3. Consider the eigenvalue/boundary value problem for y(t):− 3y" + xy = 0, y′(0) = 0, y' (√3)= 0(a) Is A = 0 an eigenvalue? If it is, calculate the corresponding eigenfunctions.(b) Determine all negative eigenvalues, A< 0, and calculate the corresponding eigenfunc-tions.Clearly show the calculations and state the reasoning justifying your conclusions.

Answered: Image /qna-images/answer/eb1d9dcf-ff03-4f34-8b5a-706a943f9be7.jpg

Answered: 3. Consider the eigenvalue/boundary…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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6. Consider the eigenvalue problem y" + ày = 0; y'(0) = 0, y(1) + y'(1) = 0. All the eigenvalues are nonnegative, so write à = a² where a 2 0. (a) Show that A = 0 is not an eigen- value. (b) Show that y = Acos ax + B sin ax satis- fies the endpoint conditions if and only if B = 0 and a is a positive root of the equation tan z = 1/z. These roots {an}° are the abscissas of the points of intersection of the curves y = tan z and y = 1/z, as indicated in Fig. 3.8.13. Thus the eigenvalues and eigenfunctions of this problem are the numbers {a;}° and the functions {cos an x}9°, re- spectively. y = 2n Зл I ly = tan z
Consider the Sturm-Liouville problem: y" + Ay /(-L) 0, -L
The eigenfunctions for y" + 2y' + Ay = y(0) = 1, y(1) = 0 are {e" sin(nπa)O {e2r sin(프플프) 1O n=1 {e-"sin(뿌)EO od In=1 4 II {e-2" sin(nπa)이 {e" sin ()O 2 n=1 .2r -:.. ( po
Solve the eigenvalue problems: Ly ₁²y = xy dr² for the two following sets of boundary conditions i. y(0)) = 0 and y(3) = 0 Ak = Yk (x) ii. y(0)) = 0 and y' (3) = 0 Ak = Yk (x) **NOTE: you must leave your answers for the eigenfunctions written in terms of X₁, rather than writing them out explicitly in terms of k, otherwise they won't be recognised as correct***
Determine the eigenvalues and corresponding eigenfunctions for the following boundary value problem: Domain xe (0,7) X" + AX = 0 with boundary conditions: X(0)=X'(7)=0
П Write the function f(x) = x (1-x) as an eigenfunction ex- pansion of the eigenfunctions corresponding to the Sturm-Liouville prob- lem y" + Xy = 0, 0 < x < y(0) = 0 y (H) = 0 πT
Find the smallest eigen value of (1+x)y" + y' + λy = 0, y(0)=0, y(1)=0
1. Find all eigenvalues An and corresponding eigenfunctions yn to the boundary value problem: y" + xy = 0 y'(0) = 0, y(5) = 0.

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