Find a fundamental set for the system a im working b

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Find a fundamental set for the system. A I’m working B I’m done C I’m stuck
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Consider the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Find a fundamental set for the system. A I’m working B I’m done C I’m stuck HINT: The eigenvalues/vectors: r 1 = - 1 1 - 1 r 2 = - 2 1 - 2
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Let’s consider the associated homogeneous equations: x 00 + 3 x 0 + 2 x = 0 which corresponds to the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Any vector-valued function of the following form is a solution: ~ x ( t ) = c 1 e - t 1 - 1 + c 2 e - 2 t 1 - 2
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Let’s consider the associated homogeneous equations: x 00 + 3 x 0 + 2 x = 0 which corresponds to the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Any vector-valued function of the following form is a solution: ~ x ( t ) = c 1 e - t 1 - 1 + c 2 e - 2 t 1 - 2 Notice that these solutions correspond to the solutions we would have gotten solving the 2nd order equation.
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Let’s consider the associated homogeneous equations: x 00 + 3 x 0 + 2 x = 0 which corresponds to the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Any vector-valued function of the following form is a solution: ~ x ( t ) = c 1 e - t 1 - 1 + c 2 e - 2 t 1 - 2 Notice that these solutions correspond to the solutions we would have gotten solving the 2nd order equation. Solutions tend to 0 as t → ∞ .
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Let’s consider the associated homogeneous equations: x 00 + 3 x 0 + 2 x = 0 which corresponds to the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Any vector-valued function of the following form is a solution: ~ x ( t ) = c 1 e - t 1 - 1 + c 2 e - 2 t 1 - 2 Notice that these solutions correspond to the solutions we would have gotten solving the 2nd order equation. Solutions tend to 0 as t → ∞ . The equilibrium x = 0 is asymptotically stable.
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Let’s consider the associated homogeneous equations: x 00 + 3 x 0 + 2 x = 0 which corresponds to the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x . Any vector-valued function of the following form is a solution: ~ x ( t ) = c 1 e - t 1 - 1 + c 2 e - 2 t 1 - 2 Notice that these solutions correspond to the solutions we would have gotten solving the 2nd order equation. Solutions tend to 0 as t → ∞ . The equilibrium x = 0 is asymptotically stable. It is called an asymptotically stable node.
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Consider the system: ~ x 0 ( t ) = 0 1 - 2 - 3 ~ x .
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Christopher Reinemann
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