A dynamic system model has state variables \(x_{1}\) and \(x_{2}\) and input \(u\). The two I/O equations are derived as

\[\ddot{x}_{1}+\dot{x}_{1}+2 x_{1}=\dot{u}+\frac{1}{3} u, \quad \ddot{x}_{2}+\dot{x}_{2}+2 x_{2}=2 u\]

Find the state-space representation for this model and subsequently the transfer matrix.