Complete the solution of Equation 6.55, [ begin{aligned} left{begin{array}{l} theta_{1}(t) theta_{2}(t) end{array} ight} & =C_{1}left{begin{array}{c} 1 1 end{array} ight} cos left(omega_{1} t-phi_{1} ight)

Complete the solution of Equation 6.55,

\[ \begin{aligned} \left\{\begin{array}{l} \theta_{1}(t) \\ \theta_{2}(t) \end{array}\right\} & =C_{1}\left\{\begin{array}{c} 1 \\ 1 \end{array}\right\} \cos \left(\omega_{1} t-\phi_{1}\right) \\ & +C_{2}\left\{\begin{array}{c} 1 \\ -1 \end{array}\right\} \cos \left(\omega_{2} t-\phi_{2}\right) \end{aligned} \]

if the initial conditions are given by

\[ \begin{array}{ll} \theta_{1}(0)=\theta_{a}, & \theta_{2}(0)=\theta_{b} \\ \dot{\theta}_{1}(0)=\Omega_{a}, & \dot{\theta}_{2}(0)=\Omega_{b} \end{array} \]