Derive Equations 6.69 and 6.70,

\[ \begin{aligned} x_{1}(t) & =e^{-a_{1} t}\left(r_{1} C_{1} \cos \omega_{d 1} t+r_{1}^{\prime} C_{2} \sin \omega_{d 1} t\right) \\ & +e^{-a_{2} t}\left(r_{2} C_{3} \cos \omega_{d 2} t+r_{2}^{\prime} C_{4} \sin \omega_{d 2} t\right) \\ x_{2}(t) & =e^{-a_{1} t}\left(C_{1} \cos \omega_{d 1} t+C_{2} \sin \omega_{d 1} t\right) \\ & +e^{-a_{2} t}\left(C_{3} \cos \omega_{d 2} t+C_{4} \sin \omega_{d 2} t\right) \end{aligned} \]