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hw02solns - Homework 2 Solution 1 Given A = 6-4 0 2 1 0 1 1...

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Homework 2 Solution 1. Given A = 6 - 4 0 2 , the eigenvalues are λ i = 6 , 2 and corresponding eigenvectors are v i = 1 0 , 1 1 . The general solution is ξ ( t ) = c 1 1 0 e 6 t + c 1 1 1 e 2 t Given ξ (0) = 2 1 = c 1 1 0 + c 2 1 1 , then c 1 = 1, c 2 = 1, and the solution is ξ ( t ) = 1 0 e 6 t + 1 1 e 2 t 2. Given A = - 3 0 0 0 - 3 1 0 1 - 3 , the eigenvalues are λ i = - 4 , - 3 , - 2 and corresponding eigenvectors are v i = 0 - 1 1 , 1 0 0 , 0 1 1 . The general solution is ξ ( t ) = c 1 0 - 1 1 e - 4 t + c 2 1 0 0 e - 3 t + c 3 0 1 1 e - 2 t Given ξ (0) = 3 2 2 = c 1 0 - 1 1 + c 2 1 0 0 + c 3 0 1 1 , then c 1 = 0, c 2 = 3, c 3 = 2, 1
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