10.4. Examples

Try these first

1. 1.

   What is the value of ∫δ(t - 1) dt over all time?

   Hint

   Think about the sifting property of the Dirac Delta Function.

2. 2.

   What is the Laplace Transform of δ(t)?

   Hint

   Consider the Dirac Delta Function centered at origin.

3. 3.

   The Dirac Delta Function is defined as δ(t - a). Which statement is true?

   • It is 0 for t = a.
   • It is infinite for t = a.
   • It integrates to 0 over all time.

   Hint

   Recall its definition!

4. 4.

   What is the result of ℒ{δ(t - 2)}?

   Hint

   Focus on the parameter a in δ(t - a) and apply it.

5. 5.

   Given a differential equation y' + 2y = δ(t - 1), find the solution y(t) and interpret its meaning.

   Hint

   Use Laplace Transforms to find Y(s) and then perform the inverse transform.

6. 6.

   A mechanical system is subject to a force modeled by f(t) = 10δ(t - 3). How does this affect the system's response?

   Hint

   Visualize the resulting movement or vibration initiated by the impulse force.