 Solution

First solve the closed linear system of differential equations for $x,y$. Its coefficient matrix is $A=\left(\begin{array}{cc}5& 3\\ 2& 0\end{array}\right)$, with characteristic polynomial $r^2-5r-6=(r-6)(r+1)$. Corresponding eigenvectors are $(3,1{)}^T$ and $(1,-2{)}^T$. The homogeneous solution is thereforeUse separate exponential particular solutions for the two forcing frequencies. Solving $(2I-A)u=(1,0{)}^T$ and $(I-A)v=(0,2{)}^T$ gives $u=(-1/6,-1/6{)}^T$ and $v=(-3/5,4/5{)}^T$. HenceThe third equation then gives $z^{\prime }=4C_1{e}^{6t}-C_2{e}^{-t}-e^{2t}/3+6e^t/5$. Integrating,The three arbitrary constants span the general solution: two arise from the first-order $x,y$ subsystem and one from integrating $z$.