-() A = Consider the system of equations 7 = AT. %3D (a) Find the eigenvalues and the associated eigenvectors of A. (b) Find a generalized eigenvector of A. (c) Find the general solution of the system.

-() A = Consider the system of equations 7 = AT. %3D (a) Find the eigenvalues and the associated eigenvectors of A. (b) Find a generalized eigenvector of A. (c) Find the general solution of the system.

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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λ1 = -2, λ2=5 V1 = (1, -2) V2 = (4, -1) Using the above eigenvalues and/or eigenvectors, find the general solution of the following system of linear differential equations, writing the solution as an equation for x and an equation for y: x ̇= 6x+4yy ̇ = − 2 x − 3 y
See the pic please (1)
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?
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1. (a) What are A, X and x' in matrix form x' = AX of the system of differential equations { FO (b) Find all eigenvalues and eigenvectors of A. (c) Find exponential solutions of the system of differential equations. (d) Verify your solutions. x₁ = 3x1 x2 = 5x1 + 2? x2
b) A system of differential equation is given as follows: x₁ = 9x₁ - 4x₂ x₂ = 8x₁ - 3x₂ where x₁ and X2 are functions of . Given that the eigenvalues of A= and 5 with the corresponding eigenvectors i) Write a matrix P that diagonalizes A. ii) Write a diagonal matrix D such that P¹ AP = D. iii) Hence, find the general solution to the system. [4=[83] are 1 and respectively.
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based on the image provided. Assisstance please.
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