8.2.2.5 Systems of differential equations

Problem: Solve the Cauchy problem for the system of linear differential equations, using the operational method: \[ \begin{array}{l} 2 x^{\prime \prime}+x-y^{\prime}=-3 \sin t, x+y^{\prime}=-\sin t, \\ x(0)=0, x^{\prime}(0)=1, y(0)=0 . \end{array} \]