hw03solns

hw03solns - Homework 4 Solution 1 7.8 7 Given A =-4...

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Homework 4 Solution 1 7.8 7. Given A = 1 - 4 4 -7 , eigenvalues are λ 1 , 2 = - 3 and corresponding eigenvectors 1 1 , 1 4 0 + k 1 1 . x = c 1 1 1 e - 3 t + c 2 1 1 te - 3 t + 1 4 0 e - 3 t . Since x (0) = 3 2 c 1 = 2 , c 2 = 4 x = e - 3 t 4t+3 4t+2 . -10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 (a) x 1 - x 2 -5 0 5 10 15 20 25 30 -25 -20 -15 -10 -5 0 5 (b) x 1 - t Figure 1: Problem 7.7 10. Given A = 3 9 -1 -3 , eigenvalues are λ 1 , 2 = 0 and corresponding eigenvectors -3 1 , -1 0 + k -3 1 . x = c 1 -3 1 + c 2 -3 1 t + -1 0 ¶‚ . Since x (0) = 2 4 c 1 = 4 , c 2 = - 14 x = 42t+2 -14t+4 . -10 -8 -6 -4 -2 0 2 4 6 8 10 -10 -8 -6 -4 -2 0 2 4 6 8 10 (a) x 1 - x 2 -10 -8 -6 -4 -2 0 2 4 6 8 10 -500 -400 -300 -200 -100 0 100 200 300 400 500 (b) x 1 - t Figure 2: Problem 7.10 1

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2 Given A = 2 1 0 0 0 0 2 1 0 0 0 0 2 1 0 0 0 0 2 1 0 0 0 0 2 , eigenvalues are λ 1 , ··· , 5 = 2 and corresponding eigenvectors and generalized eigenvectors 1 0 0 0 0 , 0 1 0 0 0 , 0 0 1 0 0 , 0 0 0 1 0 , 0 0 0 0 1 . x = e 2 t c 1 1 0 0 0 0 + c 2 t 1 0 0 0 + c 3 t 2 / 2 t 1 0 0 + c 4 t 3 / 6 t 2 / 2 t 1 0 + c 5 t
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