Consider the spring$-$mass$-$damper system defined by $($Use basic SI units$)$:

$4x^{}(t)+13x^{}(t)+100x(t)=16cos5t$

$($a$)$Compute the amplitude and the phase angle for the steady$-$state displacement response.

$($b$)$ Determine the displacement $x(t)$ for the initial conditions $x(0)=1,$ and $v(0)=0$

$($c$)$ Find the displacement response for the case of free vibration $($when the right$-$hand side is zero$)$ using the same initial conditions