Problem $1(15$ points$)$:

For a $2-$DOF system, as shown in Figure $1,$ with $m_1=m,m_2=2m,k_1=k,k_2=2k,$ and $c_1=c_2=0$ :

$1-$ Determine the equations of motion of this dynamical system.

$2-$ Find the natural frequencies and modal vectors $($or mode shapes$).$

$3-$ Determine the response of the system when $k=1000\frac{N}{m},m=20kg,$ and the initial conditions are $x_1(0)=1,x_2(0)=-1,$ and zero initial velocities.

Figure $1$