Q$1)$ Consider the following dynamical system composed of two discrete objects connected with the springs. The objects both have the same mass $m1=m2=1kg.$

$[10]$ a$)$ Derive the ODE system that governs the dynamical system given above.

$[15]$ b$)$ Considering the BCs given below solve the ODE system analytically.

BCs:

$)=(0s)=(0s)=(12s)=(12s$

$[15]$ c$)$ Discretize the ODE systems using the finite difference method. In this regard, divide the time window of $10s$ uniform time intervals $t=2s$ and discretize the ODE system and come up with a linear system of equation as a matrix $-$ vector equation.

$1$

$[60]$ d$)$ Write a Matlab code that solves numerically for the same problem.

i$)$ Use $t=0\mathrm{.}6s$ and plot the numerical solutions u$1$ & u$2$ with the analytical solutions in the same figure.

ii$)$ Use $t=0\mathrm{.}06s$ and plot the numerical solutions $u1$&$u2$ with the analytical solutions in the same figure.