**Solve the following problems**(a) **Find the eigenvalues and eigen-functions for the problem**\[ f''(x) + \lambda f(x) = 0, \quad f(0) = 0, \quad f'(2) = 0 \](b) **Discuss the orthogonality relation for the eigen-functions in part (a)**(c) **Find the general solution of the problem**\[ u_t(x, t) = 9u_{xx}(x, t), \quad 0 \leq x \leq 2, \quad t > 0 \]\[ u(0, t) = 0, \quad u_x(2, t) = 0 \]

NOTE: According to guideline answer of first three related subparts can only be given, for other…

Answered: Solve the following problems (a) Find…

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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Let x(t) x₁ (t) x' (t) x₁ (t): = = If x (0) = x₂(t): = = = ] Put the eigenvalues in ascending order when you enter x₁ (t), x₂(t) below. exp( t) + exp( x₁(t) x₂(t) -27 x₁(t) + 12x₂(t) -56 x₁(t) + 25 x₂(t) 4 be a solution to the system of differential equations: -2 exp( find x(t). t) + exp( t) t)
2. (1) Compute all eigenvalues and eigenvectors of the Cartan matrix 2 -1 C = -1 2 -1 -1 2 (2) Diagonalize C (i.e. find Q such that Q-'CQ is diagonal). (3) Is C positive definite?
For (a) and (b), 2 -1 [A] = -1 2 (a) What are the eigenvalues of A? (b) For each eigenvalue, find a nonzero eigenvector. [1/3] [1/4] (c) Suppose that the matrix B has the eigenvalues 1 and -6, with eigenvectors and respectively. Calculate Q(t).N(0)'. (Q: fundamental matrix for u’=B.u) (d) What is the solution to u'=B.u with u(0)=| ?
[1 2. (a) Find the eigenvalues and eigenvectors of B = (b) Explain why the matrix A cannot be diagonalized.
For the following matrix: a) find the eigenvalues and the eigenvectors b) check the answer by using the orthogonal property (x(i))T(x(j))=1(i≠l) c) determine the three normalized eigenvectors by using (x(i))T(x(i))=1(i≠l)
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 ()
2. Find the eigenvalues and eigenfunctions. Verify orthogonality. Show details of your work. y" - 2y' + (A + 1) y = 0, y(0) = 0, y(1) = 0

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Solve the following problems (a) Find the eigenvalues and eigen-functions for the problem ƒ” (x) + λƒ(x) = 0, (b) Discuss the orthogonality relation for the eigen-functions in part (a) f(0) = 0, f'(2) = 0