$4.($a$)$ Modify LAB$05$ex$2.$m to compute and plot the quantity E $=1/2$ mv$^2+1/2$ ky$^2$ as a function

of time. What do you observe? Is the energy conserved in this case? Include the plot.

$($b$)$ Show analytically that dE$/$dt $<mtext></mtext><mn>0</mn>$ for c $>0$ while dE$/$dt $>0$ for c

$($c$)$ Modify LAB$05$ex$2.$m to plot v vs y $($phase plot$).$ Include the plot. Comment on the

behavior of the curve in the context of the motion of the spring. Does the graph

ever get close to the origin? Why or why not?