# Solving a linear system of differential equations

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?

## 1 Answer

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.