Chapter 4.P3, Problem 6P

Consider the damped problem using the parameter values $L=1,m=1,\gamma =0.1,\text{and}k=1$. $$

(a) Use a computer to draw a direction field of the corresponding dynamical system.

(b) If you have access to computer software that is capable of solving event problems, solve for and plot the graphs of $x\left(t\right)$ and $x^{\prime }\left(t\right)$ for the following sets of initial conditions:

(i)  $x\left(0\right)=2,x^{\prime }\left(0\right)=0$

(ii) $x\left(0\right)=5,x^{\prime }\left(0\right)=0$

Give a physical explanation of why the limiting values of the trajectories as $t\to \infty$ depend on the initial conditions.

(c) Draw a phase portrait for the equivalent dynamical system.