Use modal solution approach to compute the analytical solution for the response of the $2-$degree$-$of$-$freedom damped system:

$\left[\begin{array}{cc}1& 0\\ 0& 4\end{array}\right]x^{}(t)+\left[\begin{array}{cc}540& -300\\ -300& 300\end{array}\right]x(t)=\left[\begin{array}{c}10sin35t\\ 0\end{array}\right]$

to the initial conditions of

$x_0=\left[\begin{array}{c}0\\ 0\mathrm{.}01\end{array}\right],x_0^{}=\left[\begin{array}{c}0\\ 0\end{array}\right]$

Consider the modal damping coefficients to be ${}_1=0\mathrm{.}5,$ and ${}_2=1.$