The behaviour of an unforced mechanical systemis governed by the differential equation5 2 -10x(1) = 36-9 x(t),x(0) = 11110(a) Show that the eigenvalues of the systemmatrix are 6, 3, 3 and that there is onlyone linearly independent eigenvectorcorresponding to the eigenvalue 3. Obtain theeigenvectors corresponding to the eigenvalues6 and 3 and a further generalized eigenvectorfor the eigenvalue 3.(b) Write down a generalized modal matrix Mand confirm thatAM = MJfor an appropriate Jordan matrix J.x(t) = Me M-¹x(0)obtain the solution to the given differentialequation.(c) Using the result

Problem is solved using concept of matrix eigenvalue and eigenvector.

Answered: The behaviour of an unforced mechanical…

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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How do you get to the reduced form and then get the eigen vector in this problem?
A 3×3 real matrix has three eigenvalues. One of them is λ₁ = Another is A₂ = 1 + i with eigenvector v2 - (3) = -1 with eigenvector v₁ = (a) What is the third eigenvalue and its corresponding eigenvector? (b) Find the general solution to x' = Ax, in purely real form. (c) Find the solution to x' = Ax subject to initial condition x (0) = (:). (6) 1
Consider the system of differential equations For this system, the smaller eigenvalue is (-39/10) [Note-- you may want to view a phase plane plot (right click to open in a new window).] If y'= Ay is a differential equation, how would the solution curves behave? • All of the solutions curves would converge towards 0. (Stable node) All of the solution curves would run away from 0. (Unstable node) and the larger eigenvalue is The solution curves would race towards zero and then veer away towards infinity. (Saddle) The solution curves converge to different points. The solution to the above differential equation with initial values (0) = 4, y(0) = 3 is (t)= (49/12)*e^((-9/10)*t)-(1/12)*e^((-19/10)*t) y(t) = (49/12)*e^((-9/10)*t)+(5/12)*e^((-19/10)*t) da dt dy dt = -1.4x +0.75y, = 1.66667 3.4y. (-9/10)
For a linear DDS with multiple equations, assume that A is the matrix of the system with eigenvalue c and eigenvector u. Which of the following is true? Check all that apply. X(n) = A"X(0) is the recursive equation for the DDS □X(n) = A"X(0) is the explicit solution for the DDS □X(n) = AX(n − 1) is the recursive equation for the DDS We can always compute X(n) as X(n) = c"X(0) using the eigenvalue c We can compute X(n) as X(n) = X(0) when X(0) is a multiple of
Use a calculator for: - Finding eigenvalues and eigenvectors. - Matrix multiplication still required to show how you find generalized - Matrix inversion - Row reduction -Integration5.
Solve the system of differential equations given in matrix form for the general solution. We must do this by calculating the eigenvalues and eigenvectors. Note: we get a repeated eigenvalue and must solve for the second eigenvector to get two linearly independent solutions for our general solution. Please explain how you calculate the second eigenvector
The eigenvalues of the coefficient matrix can be found by inspection or factoring. Apply the eigenvalue method to find a general solution of the system. x₁ = 7x₁ + x₂ + x3, x2 = - 13x₁ - 7x₂ -X3, X'3 = 13x₁ + 13x2 + 7x3 * A What is the general solution in matrix form? x(t) = Jiry
Let 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=
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- (1-3). Determine the eigenvalue and corres- 3. a) Given the matrix A = ponding eigenvector and generalized eigenvector of A. b) Determine the general solution of the system of linear differential equations: x₁ = I₁=I₂ x₂ = x1₁+3x₂
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