A particle moving in a planar force field has a position vector x that satisfies x'=Ax. The 2x2 matrix A has-7[1]eigenvalues 4 and 3, with corresponding eigenvectors v₁ ={at time t assuming that x(0) =x(t) =10-1-5and V₂ =Find the position of the particle

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Answered: A particle moving in a planar force…

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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a) Find the eigenvalues of A. b) Find three linearly independent solutions of x' = Ax. c) Find the exponential e^(tA) Let A = 2 -1 ا دیا 0 0 3 0 -30
Consider the vectors in C2 (sometimes called spinors) :(cos(0/2)e-ie/2 sin(8/2)et*/2 sin(8/2)e-i$/2 ) . V2 - cos(8/2)e*#/2 (i) First show that they are normalized and orthonormal. (ii) Assume that for the vector vị is an eigenvector with the corresponding eigenvalue +1 and that the vector v2 is an eigenvector with the corresponding eigenvalue -1. Apply the spectral theorem to find the corresponding 2 x 2 matrix. (iii) Since the vectors vị and v2 form an orthonormal basis in C? we can form an orthonormal basis in C4 via the Kronecker product W11 = V1 ® V1, W12 = V1 ® v2, W21 = V2 ® V1, W22 = V2 ® V2. Assume that the eigenvalue for w11 is +1, for w12 –1, for w21 -1 and for w22 +1. Apply the spectral theorem to find the corresponding 4 x 4 matrix.
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a' = with one of the phase plane direction fields below: A B D (The blue lines are the arrow shafts, and the black dots are the arrow tipa.) Note: To solve this problem, you only need to compute eiguvalus. In fact, it is enough to just compute whether the cigenvalues are real or complex and positive or negative, If you use Trace-Determinant plane, give a brief explanation about eigenvalues.
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A particle moving in a planar force field has a position vector x that satisfies x'=Ax. The 2x2 matrix A has -7 [1] eigenvalues 4 and 3, with corresponding eigenvectors v₁ = { at time t assuming that x(0) = x(t) = 10 -1 -5 and V₂ = Find the position of the particle