(Solved): 10.1 Systems of differential equations and initial value problems Use the general solution \[ \lef ...

10.1 Systems of differential equations and initial value problems Use the general solution \[ \left\{\begin{array}{l} x_{1}(t)=A e^{-1}+2 B e^{2 t} \\ x_{2}(t)=2 A e^{-t}+B e^{2 t} \end{array}\right. \] of the system \[ \left[\begin{array}{l} x_{1}^{\prime}(t) \\ x_{2}^{\prime}(t) \end{array}\right]=\left[\begin{array}{ll} 3 & -2 \\ 2 & -2 \end{array}\right]\left[\begin{array}{c} x_{1}(t) \\ x_{2}(t) \end{array}\right] \] to determine its particular solution such that \[ \left[\begin{array}{l} x_{1}(0) \\ x_{2}(0) \end{array}\right]=\left[\begin{array}{l} 1 \\ 0 \end{array}\right] \]