We want to solve the system \begin{align*} y_1'&=2y_1+3y_2\\ y_2'&=-y_1-2y_2\\ \end{align*} We start by setting up matrix $A$ and finding the eigenvectors and eigenvalues.

Pull out the eigenvalues and eigenvectors.

Create a matrix whose columns are the two independent solutions to the system.

Find the solution ${\bf y}$ that has ${\bf y}(0)=\left[\matrix{10\\-9\\}\right]$. We substitute 0 into matrix B from the previous step, then let Sage solve for the needed coefficients, $\left[\matrix{c_1\\c_2\\}\right]$. Then the actual function is $B\left[\matrix{c_1\\c_2\\}\right]$.

Plot the first 2 seconds of the trajectory, along with the lines parallel to the eigenvectors.