The eigenvalues in the exercises from 1 to 5 are all non-negative.First determine whether λ = 0 is an eigenvalue; then find the positive eigenvalues and associated eigenfunctions.1. y" + Ay = 0; y'(0) = 0, y(1) = 02. y' + λy=0; y' (0) = 0, y' (π) = 03. y" + λy = 0; y(−t) = 0, y(t) = 04. y" + y = 0; y' (−π) = 0, y' (T) = 0

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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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2. Consider the eigenvalue problem, y" + Ay = 0, for 0 < x < 7/2; y' (0) = 0, y(n/2) = 0. Find all eigenvalues and eigenfunctions for the case where the auxiliary equation has complex roots.
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PART 1 (MCQ: 8 Marks): 1. Choose the most appropriate answer. (a) The eigenvalues of the following eigenvalue problem: are given by An = 0 = (1),X = (0),x '0= xY+ „X (A) n², n=0, 1, 2, 3, - - - (C) (na)², n = 0,1, 2, 3, · - . (B) n², n=1,2,3, - - - (D) (n7)², n= 1, 2, 3, · - - (b) At x = 0, the Fourier series of x - 2, = (x)f I 5x > 0 converges to: (A) -2 (B) –1 (D) 1 (c) The solution of the following Cauchy problem: '00 > x > - u(x, 0) = 0, 4 (x, 0) = x, *00 > x > o0- is given by u(x, t) = (A) [(z+ t)? – (2 – t)*] = ct (B) (7 + t) + (x – t)] =: (C) (z +t)² + (x – t)*] = (2² + t*) 1=(9-2) -0+0)은 (a) (d) Let o(x) = x² + x, x > 0. If defines an even extension of ø to the entire real line, then for x < 0, (x) = (C) -a2 +x (A) x² +x X - „x ()
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Find the eigenvalues , and eigenfunctions y(x) for the given boundary-value problem. (Give your answers in terms of n, making sure that each value of n corresponds to a unique eigenvalue.) y" + 2y +(2+1)y = 0, y(0) = 0, y(9) = 0 Yn(x) = 2 ²n² 18 Need Help? Read It n = 1, 2, 3, ... n = 1, 2, 3,...

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The eigenvalues in the exercises from 1 to 5 are First determine whether λ = 0 is an eigenvalue; X 1. y" + xy = 0; y'(0) = 0, y(1) = 0 Jac -0. 2(0) 0 (π) 2 2