(Solved): (25 points) Let \( x(t)=\left[\begin{array}{c}x_{1}(t) \\ x_{2}(t)\end{array}\right] \) be a soluti ...

(25 points) Let \( x(t)=\left[\begin{array}{c}x_{1}(t) \\ x_{2}(t)\end{array}\right] \) be a solution to the system of differential equations: \[ \begin{array}{l} x_{1}^{\prime}(t)=17 x_{1}(t)+18 x_{2}(t) \\ x_{2}^{\prime}(t)=-12 x_{1}(t)-13 x_{2}(t) \\ \text { If } x(0)=\left[\begin{array}{c} 2 \\ 1 \end{array}\right] \text {, find } x(t) . \end{array} \] Put the eigenvalues in ascending order when you enter \( x_{1}(t), x_{2}(t) \) below, \[ \begin{array}{lll} x_{1}(t)= & \exp (\quad t)+ \\ x_{2}(t)= & \exp ( & \exp ( \end{array} \] t) t)