Section 2. State Space

The forced oscillator is modeled with a non-autonomous equation. Solution trajectories are best viewed in state space.

1. Consider the driven IVP

y'' + 4y = cos(1.8 t)  , y(0) = 2 , y'(0) = -3

Obtain the solution, call it soln and convert it into a function g using unapply.

a. Plot the solution curve over the interval t = 0..120. What you witness in the plot is the phenomenon called "beats". The output pulsates like this when the driver frequency is very close to the natural frequency of the system and the system is lightly damped.

b. Obtain the phase plane trajectory for this system. Use the same time interval.

c. Obtain the state space trajectory for this system. Use the same time interval. Add the equations numpoints=800, axes=framed, orientation=[-60,70], labels=["Position","Velocity","Time"] .

2.* Damp the system slightly by changing the IVP to the following

y'' + 0.1y' + 4y = cos(1.8 t)  , y(0) = 2 , y'(0) = -3

Obtain the solution, call it soln and convert it into a function g using unapply. Suppress the output for soln and the definition of g and then enter

evalf[3]( 'g(t)' = g(t) )

to see a nice looking representation of the solution formula.

a. Based upon the solution formula determine the time constant for the beats. That is, how long will it take (approximately) for the beats to disappear from the solution curve as it settles down to its steady-state mode?

b. Plot the solution curve to verify your answer to part a.

c. Use g to obtain the phase plane and state space trajectories for this system.

3. Define the procedure called DEsystem in Part 4 Section 2 of the manual (page 81). Apply it to the differential equation in Exercise 1 of Section 1 (this Part) and use DEplot to obtain the direction field and the solution trajectory. (The direction field can be drawn in phase space because the equation is autonomous.)

4. Apply DEsystem it to the differential equation in Exercise 1 in this section and then use DEplot to obtain the solution trajectory. (The direction field cannot be drawn in phase space because the equation is not autonomous.)

5.* Apply DEsystem it to the differential equation in Exercise 2 in this section and use DEplot to obtain the solution trajectory.

6.* Continuing 5. Apply DEplot3d to obtain the state space trajectory for the IVP in Exercise 2.

7. Use dsolve/numeric to obtain a numeric solution for the IVP in Exercise 2. Apply odeplot to sketch the time series for position and the time series for velocity. Then obtain the phase plane trajectory and the state space trajectory using odeplot.

8. Obtain a nice display of numerical solutions to the IVP in Exercise 2 by using dsolve/numeric with the

output=array( list )

option. Make the display even nicer by suppressing the output and then entering

evalf[4](%)