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Question

# Construct the general solution of x'=Ax involving complex eigenfunctions and then obtain the general real solution. Describe the shapes of typical trajectories.$A=\left[ \begin{array}{rrr}{-8} & {-12} & {-6} \\ {2} & {1} & {2} \\ {7} & {12} & {5}\end{array}\right]$

Solution

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$A=\begin{bmatrix}-8&-12&-6\\2&1&2\\7&12&5\end{bmatrix}$

$\lambda_1=-2, \lambda_2=1, \lambda_3=-1$

For the given matrix A, the eigenvalues are $\lambda_1, \lambda_2, \lambda_3$.

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