Q1) Consider the problem y " +2y= 0, y' (0) = 0, y(1) = 0. a) Find the eigenvalues and the eigenfunctions. b) Expand f(x)=x using the above functions.
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![Q1) Consider the problem y"+y= 0, y (0) = 0, y(1) = 0.
a) Find the eigenvalues and the eigenfunctions.
b) Expand f(x)=x using the above functions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Feed1351b-686f-4d8e-80ba-a8bc3ffcf56d%2Fc48f23a4-eb19-4621-8b18-2f9ce33fb516%2Fy851hz5_processed.png&w=3840&q=75)
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- 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 πTWhat are the eigenvalues and eigenfunctions: x′′ + λx = 0, x(1) = x(3), x′(1) = x′(3) - Definey by x(z) = y((z−2)π) for 1 ≤ z ≤ 3. Show that y(−π) = y(π) and y′(−π) = y′(π) - Substitute into the equation to get y′′((z−2)π) + (λ/π2) y((z−2)π), for 1 ≤ z ≤ 3 - Use the change of variable t = (z − 2)π to show that the above equation has a non-zero solution if and only if either λ = k2π2 for some integer k ≥ 1 or λ = 0 and the solutions (eigenfunctions) are given by cos(kt), sin(kt) and 1 for −π ≤ t ≤ π. - Plug back t = (z − 2)π to find the eigenfunctions and eigenvalues of original equation1. Find all eigenvalues An and corresponding eigenfunctions yn to the boundary value problem: y" + xy = 0 y'(0) = 0, y(5) = 0.
- Find the eigenvalues and eigen functions of the Strum-Liouville problem u" + Au =0, 0sxQ.2 Use eigenvalues and eigenvectors to solve : 2 X' = -1 8 |х, X(0) —| - 2 2Let A be a 2 x 2 matrix with eigenvalues X₁ = -3 and >₂ = -1 and [1] an H Let x(t) be the position of a particle at time t. Suppose we solved the initial corresponding eigenvectors V₁ = value problem x = Ax, x(0) = [3] -ce + c₂e-t x(t) = = [2(t)] = [ qe * +ge + -3t Search and C2= -3t and V2 = to obtain then C1=I 5. Find all eigenvalues and eigenfunctions to the boundary value problem. Show all necessary work: y" + Ay = 0, y'(0) = 0, y' (L) = 0When solving the Sturm-Liouville boundary value problem y" - Ay=0, 03- Solve the problem P.D.E. u, = uxx 0PART 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 ()Given that A = (14) has eigenvalues -1 and 3 with 3-eigenvector equal to [H] 2 -1 A = ^-636369* and -1-eigenvector equal to 2 , fill in the blank:Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,Advanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, IncorporatedNumerical Methods for EngineersAdvanced MathISBN:9780073397924Author:Steven C. Chapra Dr., Raymond P. CanalePublisher:McGraw-Hill EducationIntroductory Mathematics for Engineering Applicat…Advanced MathISBN:9781118141809Author:Nathan KlingbeilPublisher:WILEYMathematics For Machine TechnologyAdvanced MathISBN:9781337798310Author:Peterson, John.Publisher:Cengage Learning,