= Solution
First solve the closed <linear system of differential equations> for $x,y$. Its coefficient matrix is $A=\begin{pmatrix}5&3\\2&0\end{pmatrix}$, 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 therefore
$$
\begin{pmatrix}x_h\\y_h\end{pmatrix}
=C_1e^{6t}\begin{pmatrix}3\\1\end{pmatrix}
+C_2e^{-t}\begin{pmatrix}1\\-2\end{pmatrix}.
$$
Use 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 $v=(-3/5,4/5)^T$. Hence
$$
\boxed{x=3C_1e^{6t}+C_2e^{-t}-\frac16e^{2t}-\frac35e^t,\qquad
y=C_1e^{6t}-2C_2e^{-t}-\frac16e^{2t}+\frac45e^t.}
$$
The third equation then gives $z'=4C_1e^{6t}-C_2e^{-t}-e^{2t}/3+6e^t/5$. Integrating,
$$
\boxed{z=\frac23C_1e^{6t}+C_2e^{-t}-\frac16e^{2t}+\frac65e^t+C_3.}
$$
The three arbitrary constants span the general solution: two arise from the first-order $x,y$ subsystem and one from integrating $z$.
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