A 2 × 2 real matrix A with repeated eigenvalues satisfies the relation0A¹8-191[2]. then the general solution to the linear system y' = Ay has theVform"and hasy(t) = c₁ [2]C1[1²2] + 0₂ [(1 + 2) 6-21]e-2t-2t+2)e-2ty(t) = a₁ [ ²2 ] + 0₂ [(2C1-2ty(t) = a₁ [eu]C1-2ta corresponding generalized eigenvector+ C2C₂te-2t(2t + 1)e-2tte-2t2te-2t°-[]+[]y(t) = C₁-2t2]2e-2t

Given, The relation satisfies by the real matrix 2×2 with repeated eigenvalues is given by A01=0-2

A 2 × 2 real matrix A with repeated eigenvalues satisfies the relation [i] = [-2], and has a corresponding generalized eigenvector 1 [2]. then the general solution to the linear system y' = Ay has the V= form O y(t) = 0₁ [ ²2 ] + 0² [( 1 + 2) C1 y(t) = c₁ [ ²2 ] + c² [ (21 C1 te-2t (2t + 1)e-2t -2t te-2t y(t) = C₁ []+[²] C₂ y(t) = ₁₁₁ C1 -2t e-2t +2)e-2t 0 -2t + C₂ 2te-2t -2t e 2e-2t

A 2 × 2 real matrix A with repeated eigenvalues satisfies the relation [i] = [-2], and has a corresponding generalized eigenvector 1 [2]. then the general solution to the linear system y' = Ay has the V= form O y(t) = 0₁ [ ²2 ] + 0² [( 1 + 2) C1 y(t) = c₁ [ ²2 ] + c² [ (21 C1 te-2t (2t + 1)e-2t -2t te-2t y(t) = C₁ []+[²] C₂ y(t) = ₁₁₁ C1 -2t e-2t +2)e-2t 0 -2t + C₂ 2te-2t -2t e 2e-2t

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.1: Eigenvalues And Eigenvectors

Problem 80E

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