8.2.2.4 Systems of differential equations

Problem: Solve the system of differential equations, using the operational method: \[ \left\{\begin{array}{l} y^{\prime}+5 y-2 x=e^{t} \\ x^{\prime}-y+6 x=e^{-2 t}, \quad x(0)=0, \quad y(0)=0 . \end{array}\right. \]