Collins Carroll

12/07/2023 · Elementary School

\( A=\left[\begin{array}{cc}-6 & 3 \\ -2 & -1\end{array}\right] \) has the following eigenpairs. \[ \left(r_{1}=-4, \mathbf{c}_{1}=\left[\begin{array}{l}3 \\ 2\end{array}\right]\right)\left(r_{2}=-3, \mathbf{c}_{2}=\left[\begin{array}{l}1 \\ 1\end{array}\right]\right) \] Find the general solution to the following system of differential equations. \[ \begin{array}{l}y_{1}^{\prime}=-6 y_{1}+3 y_{2} \\ y_{2}^{\prime}=-2 y_{1}-y_{2}\end{array} \] \[ \begin{array}{l}\mathbf{y}=k_{1}\left[\begin{array}{r}\square \\ \square\end{array}\right] e^{\square} t+k_{2}\left[\begin{array}{l}\square \\ \square\end{array}\right]\end{array} \]