For the system shown,

$m_1=1kg,m_2=4kg,k_1=10\frac{N}{m},k_2=2\frac{N}{m},$ and $k_3=10$N$/$m

$($a$)$Write the equations of motion in matrix form

$($b$)$ Find the eigenvalues and eigenvectors for the system

$($c$)$ Draw clearly the mode shapes displaying the relative motion of the masses.

$($d$)$ Identify any nodes for the system.