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Unformatted text preview: values Î» = 3 Â± 2 i with corresponding eigenvectors Â± 1 1 Â² Â± i Â± 2 Â² . We can write down the general solution to the system of diï¬€erential equations: it is ~x ( t ) = c 1 e 3 t Â³Â± 1 1 Â² cos 2 tÂ± 2 Â² sin 2 t Â´ + c 2 e 3 t Â³Â± 1 1 Â² sin 2 t + Â± 2 Â² cos 2 t Â´ . Using the initial condition ~x (0) = Â± 2 4 Â² , we have that Â± 2 4 Â² = c 1 Â± 1 1 Â² + c 2 Â± 2 Â² . Writing this out as equations, we get c 1 = 2 c 1 + 2 c 2 = 4 1 We thus have c 1 = 2 and c 2 = 1, so our solution is ~x ( t ) = 2 e 3 t Â±Â² 1 1 Â³ cos 2 tÂ² 2 Â³ sin 2 t Â´ + e 3 t Â±Â² 1 1 Â³ sin 2 t + Â² 2 Â³ cos 2 t Â´ = e 3 t Â±Â² 2 4 Â³ cos 2 t + Â² 13 Â³ sin 2 t Â´ . 2...
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 Spring '07
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 Equations, initial condition

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