Let y' ==Ay be a system of differential equations where A=The matrix has spectrum \(A) = {2} and c =The vector d =5[]3[4] =Y2kie-2-15[5]What is the general solution to the system of differential equations?tsatisfies the equation (A — rI)d = C.Ex: 62 5[69]0 2is an eigenvector of A corresponding to r = 2.+k₂ tett] + ₁₁ [])

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Answered: Let y' = Ay be a system of differential…

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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Part: b:Diagonalize the following matrix if possible whose eigenvalues are given to be 0, 1, 1. -1 -1] A= 1 2 1 -1 Using this diagonalization, write three matrices product in its simplest form for Al0. Is it same for any power Akfor this special matrix A. Part: c: Suppose the rabbit-wolf population (rw) are governed by the following coupled differential equation dr = ar + bw dt dw = cr + dw dt Where a, b, c and d are some unknown constants. If this coupled differential equation is written as a vector differential equation as du = Au dt [a b] where A=" lc dl 1 = |" and u= And given that eigenvalues of A are -3 and -1 with corresponding eigenvectors vi= E&v= find the rabbit and wolf population i.e the functions r(t) and w(t) at any time t if initial rabbit population ro is 2 and wolf population wo is 3?
(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.
1
Consider the system of linear differential equations:
First digit is 2 last digit is 8
A system of ordinary differential equations has 2 x 2 state matrix with eigenvalues and corresponding eigenvectors (eigenpairs) A1 = 5, v1 = and A2 = -6, Based on the given eigenpairs write down the general solution to the associated system of differential equations. The solution must be written as a |¤1(t) single vector like a =
lloa [Ur ses SEC::116 Question 9 Find the general solution to the system x' = Ax where A is the given matrix. -2 -1 1 A= -1 -9 3 -1 -2 -4 Hint: -6 is an eigenvalue.
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)

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