# 582hw3 - EEE 582 Problem 5.2a-b HW 3 SOLUTIONS 1 The...

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EEE 582 HW # 3 SOLUTIONS Problem 5.2a-b 1. The characteristic polynomial is given by det( ¸I ¡ A )=0 , ¸ 2 +2 ¸ + 1 = 0 hence ¸ 1 ; 2 = ¡ 1. The exponential matrix is given by: e At = ¯ 0 ( t ) I + ¯ 1 ( t ) A .Th efunc t ion s ¯ 0 and ¯ 1 are given by: e ¸t = ¯ 0 + ¸¯ 1 te ¸t = ¯ 1 repeated eigenvalues so ¯ 1 = te ¡ t and ¯ 0 = e ¡ t + te ¡ t .Th en e At = · t +1 t ¡ t 1 ¡ t ¸ e ¡ t 2. The characteristic polynomial is given by det( ¸I ¡ A )=0,( ¸ +1) 3 = 0 hence ¸ 1 ; 2 ; 3 = ¡ 1. The exponential matrix is given by: e At = ¯ 0 ( t ) I + ¯ 1 ( t ) A + ¯ 2 ( t ) A 2 .Thefun c t ion s ¯ 0 , ¯ 1 and ¯ 2 are given by: e ¸t = ¯ 0 + ¸¯ 1 + ¸ 2 ¯ 2 te ¸t = ¯ 1 +2 ¯ 2 ¸ repeated eigenvalues t 2 e ¸t =2 ¯ 2 so ¯ 2 = t 2 2 e ¡ t , ¯ 1 = te ¡ t + t 2 e ¡ t and ¯ 0 = e ¡ t + te ¡ t + t 2 2 e ¡ t .Th en e At = 2 4 e ¡ t 00 0 e ¡ t 0 te ¡ t 0 e ¡ t 3 5 Problem 5.7 Tak ingder ivat ivesonboths ides d dt · A Z t 0 e ¸ =

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582hw3 - EEE 582 Problem 5.2a-b HW 3 SOLUTIONS 1 The...

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