Suppose that the matrix A has repeated eigenvalue with the following eigenvector and generalized eigenvector: $\lambda = 2$ with eigenvector $\vec{v} = \begin{bmatrix} 3 \\ 4 \end{bmatrix}$ and generalized eigenvector $\vec{w} = \begin{bmatrix} 1 \\ -3 \end{bmatrix}$ Write the solution to the linear system $\vec{r'} = A\vec{r}$ in the following forms. In eigenvalue/eigenvector form: $\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = c_1$ help (matrices) In fundamental matrix form: $\begin{bmatrix} x(t) \\ y(t) \end{bmatrix} = $ help (formulas) help (matrices) As two equations: (write "c1" and "c2" for $c_1$ and $c_2$) x(t) = y(t) = help (formulas) help (formulas) Note: If you are feeling adventurous you could use other eigenvectors like $4\vec{v}$ and other generalized eigenvectors like $\vec{w} - 3\vec{v}$.
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Step 1: The general solution to the system $\vec{r}' = A\vec{r}$ is given by $\vec{r}(t) = c_1 e^{\lambda t} \vec{v} + c_2 (t\vec{v} + \vec{w})e^{\lambda t}$. Show more…
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Suppose that the matrix A has the following eigenvalues and eigenvectors: λ1 = 2 with eigenvector [8, 1] and λ2 = 3 with eigenvector [3, 2]. Write the general real solution for the linear system T = Ar, in the following forms: A. In eigenvalue-eigenvector form: [r] = C1 * [8, 1] * e^(2t) + C2 * [3, 2] * e^(3t) In fundamental matrix form: [r] = [8e^(2t), 3e^(3t)] * C1 + [e^(2t), 2e^(3t)] * C2 C. As two equations: (write "c1" and "c2" for C1 and C2) y(t) = 8c1 * e^(2t) + 3c2 * e^(3t) z(t) = c1 * e^(2t) + 2c2 * e^(3t)
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