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Consider the system of coupled differential equations:$$\begin{cases}\dfrac{dx}{dt} = 5x + 4y \\ \dfrac{dy}{dt} = x + 2y\end{cases}$$
  1. Write the system in the matrix form \(\dot{\mathbf{x}} = \mathbf{A}\mathbf{x}\).
  2. Find the eigenvalues \(\lambda_1\) and \(\lambda_2\) of the matrix \(\mathbf{A}\).
  3. Find the eigenvector \(\mathbf{v}_1\) corresponding to \(\lambda_1\) and the eigenvector \(\mathbf{v}_2\) corresponding to \(\lambda_2\).
  4. Hence, write the general solution for the system.

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