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solutions6ac - Math 33b, Quiz 6ac, March 4, 2008 Name: UCLA...

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Math 33b, Quiz 6ac, March 4, 2008 Name: UCLA ID: 1. Find the solution, with initial condition ~x (0) = ± 2 3 ² , to the system of differential equations d~x dt = ± - 1 - 1 4 - 5 ² ~x. Solution. The characteristic equation of the matrix is det( A - λI ) = det ± - 1 - λ - 1 4 - 5 - λ ² , which is λ 2 + 6 λ + 9. The roots are λ = - 3 , - 3, so there is only one eigenvalue λ = - 3. We attempt to find eigenvectors for λ = - 3. Solving ( A - ( - 3) I ) ~v = ~ 0, we have ± 2 - 1 4 - 2 ²± v 1 v 2 ² = ± 0 0 ² . Writing them out as equations, we have 2 v 1 - v 2 = 0 4 v 1 - 2 v 2 = 0 As usual, the two equations are multiples of each other, so we need only find a nonzero solution to 2 v 1 - v 2 = 0. If we choose v 1 = 1, then v 1 = 2 and so ~v = ± 1 2 ² . To write down the solution to the system of differential equations, we must find a pseudoeigenvector ~w such that ( A - ( - 3) I ) ~w = ~v . Writing everything out, we want ± 2 - 1 4 - 2 ²± w 1 w 2 ² = ± 1 2 ² . Writing these out, we have 2 w 1 - w 2 = 1 4 w 1 - 2 w
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solutions6ac - Math 33b, Quiz 6ac, March 4, 2008 Name: UCLA...

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