APM3711 Assignment 3 (COMPLETE ANSWERS) 2024 - DUE 13 June 2024

APM3711 Assignment 3 (COMPLETE ANSWERS) 2024 - DUE 13 June 2024 ;100 % TRUSTED workings, explanations and solutions. For assistance call or W.h.a.t.s.a.p.p us on ...(.+.2.5.4.7.7.9.5.4.0.1.3.2)........... 1. Solve the boundary–value problem y 00 + x 2 y 0 − 4xy = 2x 3 + 6x 2 − 2, y (0)= 0, y (1)= 2 by using the shooting method. Use the modified Euler method (with only one correction at each step), and take h = 0.2. Start with an initial slope of y 0 (0) = 1.9 as a first attempt a nd y 0 (0) = 2 .1as a second attempt. Then interpolate. Compare the result with the analytical solution y = x 4 − x 2 + 2x . (7) 2. (a) Define what is meant by the eigenvalues and eigenvectors of a matrixA. If the matrix A is A =   2 0 1 − 22 − 3 10 − 12 0 9   , (b) find the characteristic polynomial, (c) find the eigenvalues and eigenvectors. (d) Start with the approximate eigenvector (1,1,1) and use the power method to estimate the dominant eigenvalue by iterating 4 times. (e) Use the power method to find the smallest absolute eigenvalue of A. (f) Write a program which applies the power method to a given matrix in (d) and (e) above. (25) 3. Consider the following boundary–value problem: y 00= 2x 2 y 0 + xy + 2, 1 ≤ x ≤ 4. Taking h = 1, set up the set of equations required to solve the problem by the finite difference method in each of the following cases of boundary conditions: (a) y (1) =− 1, y (4) = 4; (b) y 0 (1) = 2, y 0 (4) = 0; (c) y 0 (1) = y (1), y 0 (4) = − 2y (4). (Do not solve the equations!). 16 (15)