The mass$-$spring setting, as shown on Figure $1,$ is considered as our system.

Figure $1$: Mass $$ spring setting

F $=2$sin$(5$t$)$

F $=-2$

F$=-2$ exp$(9-$t$)$

F $=0$

At time t$=0$ the mass is at rest and spring is compressed. Find the motion x$($t$)$ and plot it for for the forcing F$($t$)$ is varying as shown in the figure.

$($a$)$ Without any math, how would you predict the motion of the mass m$?$ Draw a rough plot depicting the motion x$($t$)$ with respect to time. Give intuitive reasons for your plot.

$($b$)$ Using the Newton$$s second law and the free body diagram of the system, obtain the governing equation of motion.

$($c$)$ Solve part $($b$)$ using any Ordinary Differential Equation $($ODE$)$ solution method to obtain an analytical expression for x$($t$).$

$($d$)$ using matlap Plot x$($t$)$ versus time t$.$ Compare this result with part $($a$).$ Justify your comparison; but please don$$t redraw part $($a$)$ at this point $.$