For each system shown below, write the differential equation of motion and provide the expression for the natural period. Note that $m$ is mass, $k_1$ and $k_2$ are spring constants, $f(t)$ is external force, and $y(t)$ is support displacement.

$v(t)$

c$)$

Note the $\text{'}$ 'isplacement $y(t)$ of the left$-$hand support.