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Given that $v_1 = \begin{bmatrix}1&1\end{bmatrix}$ and $v_2 = \begin{bmatrix}2 &1\end{bmatrix}$ are eigenvectors of the matrix $$ \begin{bmatrix}-1&-2\\1&-4\end{bmatrix} $$ which is a $2\times 2$ matrix.

Find the solution to the linear system of differential equations \begin{align*} x' &= -x - 2y\\ y' &= x - 4y \end{align*} satisfying the initial conditions $x(0)=7$ and $y(0)=5$.

So I already found the eigenvalues, $-3$ and $-2$ and I know that you need to plug the eigenvalues into the matrix you get from doing $\det(It - A)$ but I'm not sure where to go from there in terms of making it into an equation?

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We can write the solution to the system as

$$X(t) = \begin{bmatrix} x(t) \\ y(t)\end{bmatrix} = c_1 e^{\lambda_1 t} v_1 + c_2 e^{\lambda_2 t} v_2$$

From the given information, we have

$$X(t) = c_1 e^{-3 t}\begin{bmatrix} 1 \\ 1 \end{bmatrix} + c_2 e^{-2 t}\begin{bmatrix} 2 \\ 1 \end{bmatrix}$$

Now, use the initial conditions to solve for $c_1$ and $c_2$. You can see examples here.

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