Homework Problems

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Problem 1. Find the following Laplace transforms using Maple. Describe briefly what formulas and theorems you would use if you had to find the transforms by hand. (For simplicity's sake we denote the transform by "L" rather than by the script L.)

(a) , where w and d are constants.

(b) , where a , b are constants.

(c)

(d) .

( As in our text, we denote by the unit step function at t=3. The unit step function is also called the Heaviside function. Use the proper Maple syntax to enter the Heaviside function.)

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Problem 2. Find the following inverse transforms using Maple. Describe briefly what formulas and methods you would use if you had to find the inverse tranforms by hand.

(a)

(b)

(c) , where w is a constant. (This is an example of Maple giving you an answer that looks different from the answer you may expect. You have to apply combine(...,trig) to the result to see a simpler answer.)

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Problem 3. Solve the following initial value problems. Plot the solutions and explain their physical meaning assuming that both equations describe a mass-spring system with an external driving force given by the right-hand side.

(a) y''(t) + 16y(t) = 4 (t-1) , y(0)=0, y'(0)=0.

By we denote the Dirac impulse. Use the proper Maple syntax to enter the Dirac impulse!

(b) y''(t) + y(t) = p(t) , y(0)=0 , y'(0)=0, where p(t)=1 for 1 < t < 2, p(t)=0 otherwise.

Remember to enter properly the discontinuous right-hand side p(t)! The solution provided by Maple simplifies if you apply to it the command combine(...,trig).

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