Forced Oscillations We study forced oscillations because they describe the motion and dynamics of molecules and atoms. We use the model of the spring$-$mass system to describe the oscillatory behavior observed at the molecular level. To this end, consider the following IVP $($initial$-$value problem$).$

$u^{\text{'}\text{'}}+u=3cos(t),u(0)=0,u^{\text{'}}(0)=0.$

In this problem, ${}_0=1$ is the natural frequency of the homogenous equation.

$($a$)$ Find the solution, using dsolve for $=0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}7,0\mathrm{.}8,0\mathrm{.}9.$ Plot the solution curves. What happens as ${}_0?$ To do this, you may want to use a 'for' loop as you did in the previous problem:

clear all;

Use dsolve to solve the equation, using a variable for omega, such as w$.$ This process is similar to what you did in $1($c$)$ of MATLAB Assignment $1.($b$)$ Plot the solution for omega$=0\mathrm{.}9$ on a longer interval. What do you notice?$($c$)$ The formula found in part $($a$)$ is invalid when omega$=1($why$?).$ Find and plot the solution for omega$=1($using MATLAB$).$ What seems to happen? Why might you expect this? $($You may have to search for some information on forced vibrations in your textbook$).$ Use mat lab to answer