Let A be a 2 x 2 matrix with eigenvalues X₁ = -3 and >₂ = -1and[1] anHLet x(t) be the position of a particle at time t. Suppose we solvedthe initialcorresponding eigenvectors V₁ =value problem x = Ax, x(0) = [3]-ce+ c₂e-tx(t) == [2(t)] = [ qe * +ge +-3tSearchand C2=-3tand V2 =to obtainthen C1=

Given that the solution to the initial value problem , is .The objective is to find the value of…

Answered: Let A be a 2 x 2 matrix with…

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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(b) Determine the eigenvalues and eigenvectors of the matrix Hence, write down the associated modal matrix P and diagonal spectrum matrix D, and use these to solve the following system of differential equations * = 4x-2y j =x+y
( 3). Consider the system of equations x' = Ax where A = The eigenvalues for the matrix A are d1 = 2, A2 = 3. Find the corresponding eigenvectors vị and v2.
35. (a) Show that the eigenvalues of the 2 X 2 matrix a b A = c d] с а are the solutions of the quadratic equation 12 – tr(A)A + det A = 0, where tr(A) is the trace of A. (See page 162.) (b) Show that the eigenvalues of the matrix A in part (a) are A = {(a + d ± V(a – d)? + 4bc) (c) Show that the trace and determinant of the matrix A in part (a) are given by tr(A) - λι + λ and det A = λιλε where A, and A2 are the eigenvalues of A. 36. Consider again the matrix A in Exercise 35. Give conditions on a, b, c, and d such that A has (a) two distinct real eigenvalues, (b) one real eigenvalue, and (c) no real eigenvalues.
Suppose that a 2 x 2 matrix A with real entries has an eigenvalue X = 1 + 3i with corresponding 4 i [12] eigenvector 1 + 2i Which of the following is NOT a solution of the linear system y' = Ay? y(t) = et y(t) = et y(t) = et y(t) = et 4 cos(3t) sin(3t) [cos(3t) + 2 sin(3t). 4 cos(3t) + sin(3t) cos(3t) 2 sin (3t) - cos(3t) + 4 sin(3t) 2 cos(3t) + sin(3t) cos (3t) - 4 sin (3t) -2 cos(3t) - sin(3t)
Q3. (a) If Ax = 2 x, determine the eigenvalues and the corresponding eigenvectors for A= G ) 2 3. Hence, write down the associated modal matrix P and diagonal matrix D, and use these values to solve the following system differential equations: *1 = X1 + 4 x2 X2 = 2 x1 + 3 x2 Given that when t = 0, x1 = 0 and x2 = 2. (b) Use the 4th order Runge Kutta method to solve the differential equation: dy e* y dx for values of x 0 (0.2) 0.4 given that y 1 when x 0. %3D Give your answers correct to 5 decimal places. Obtain the analytical solution of the differential equation and compare the analytical solution when x = 0.4 with the values obtained using Runge Kutta,

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