For the system shown below, complete the following:

$(1)$ Draw free$-$body diagram of each mass, find the equation of motion by applying Newton's Law of Motion in terms of $x_1$ and $x_2,$ and put the equations in matrix form. Identify the mass matrix and stiffness matrix.

$(2)$ Solve the eigenvalue problem and find the natural frequencies and mode shapes when $m_1=20kg,m_2=30kg,k_1=1000\frac{(N)}{(m)},$ and $k_2=1500\frac{(N)}{(m)}.$

$(3)$ If the system is disturbed by initial displacements vec$(x)(0)={\left[\begin{array}{cc}1& -2\end{array}\right]}^T,$ with zero initial velocity, use the orthogonality property of the mode shapes to find the free vibration response. See class notes. Do not solve simultaneous equations.