Section 5. The Laplace Transform

The inttrans package contains procedures for calculating the Laplace transform and the inverse Laplace transform. Piecewise defined functions can be defined using the Heaviside function (unit step function in some texts). Dirac delta functions in the driver can be handled via Laplace transforms.

Use Maple to do the following problems. Load the inttrans package first via  with(inttrans) . Define aliases for the Heaviside function and the Dirac delta via  alias( H=Heaviside, delta=Dirac ) .

1. Obtain the Laplace transform of the following functions.

t cos(3 t)  ,  exp(-t) sin(2 t)  ,  H(t - Pi) (t + cos(t))  ,  sin(t) + delta(t - 3 Pi)  ,  

piecewise( t < 3, 0, t < 5, t - 3 , 0 )

2. Obtain the inverse Laplace transform of the following functions.

s/(s^2-4)  ,  s*exp(-2*s)/(s^2-4)  ,  1/(s^2+4*s-5)  ,  1/(s^2+4*s-4)  ,  exp(-3*s)/(s^4)

3. Use the convert procedure to convert the piecewise function in Exercise 1 to Heaviside form and obtain the Laplace transform.

Note. Assuming that you have defined the aliases as indicated above, name the piecewise function Joe, and enter

Jose := convert(Joe,H) .

Plot Joe and Jose separately to check that they are the same. Compare the Laplace transforms of Joe and Jose. Are they also the same?

4. Consider the following initial value problem

y'' + y' + 9 y = H(t - 2 Pi)  ,  y(0) = 0 , y'(0) = 1

a. Obtain the solution using dsolve. Plot it for t = 0..40  and explain the behavior of the solution curve.

b. Obtain the solution using dsolve/method=laplace. How does the solution formula compare to the formula obtained in part a?

5. Consider the following initial value problem

y'' + y' + 9 y = H(t - 2 Pi) - H(t - 5 Pi)  ,  y(0) = 0 , y'(0) = 1

a. Obtain the solution using dsolve. Plot it for t = 0..40  and explain the behavior of the solution curve.

b. Obtain the solution using dsolve/method=laplace. How does the solution formula compare to the formula obtained in part a?

6.* Consider the following initial value problem

y'' + y' + 9 y = H(t - 2 Pi) - H(t - 5 Pi) + 2 Dirac(t - 9 Pi)  ,  y(0) = 0 , y'(0) = 1

a. Obtain the solution using dsolve. Plot it for t = 0..40  and explain the behavior of the solution curve.

b. Obtain the solution using dsolve/method=laplace. How does the solution formula compare to the formula obtained in part a?

7.* Use unapply to convert the solution to 6 a into a function, g. Use g to plot the phase plane trajectory and as well as the state space trajectory for the system. Use the following options in the spacecurve procedure for the state space trajectory

axes=boxed, orientation=[-60,70], color=red, numpoints=400, labels=["Position","Velocity","Time"]

8.* Continuing 7. Use the entry

plots[animate]( plot, [ [g(t),D(g)(t),t=0..T] ], T=0..40, frames=82);

to plot an animation of the phase space trajectory. Click on the plot and use the video controls in the context bar to play the animation and also to view it one frame at a time. Compare the trajectory movement to the time series for position.

Note. The slide control can also be used to control the display of the frames. Assuming t is measured in seconds, you can reduce the frame rate to 2 frames per second to view the trajectory in "real time".