1.8.1. Example 3

Try these first

1. 1.

   Find the Laplace Transform of e^(2t) sin(t).

   Hint

   Use the theorem to shift based on the coefficient of the exponent.

2. 2.

   What do you change in the Laplace domain when you have e^(at)?

   Hint

   Remember the theorem statement!

3. 3.

   What does the First Shifting Theorem allow you to do?

   • Transform functions without any shifts
   • Shift the s variable based on an exponential term
   • Only apply to periodic functions

   Hint

   Think about how exponential functions interact with transforms.

4. 4.

   True or False: Multiplying a function by e^(3t) shifts to the right in the frequency domain.

   • True
   • False

   Hint

   Recall the direction of the shift when applying the theorem.

5. 5.

   Demonstrate the application of the First Shifting Theorem in solving a second-order differential equation involving e^(3t) sin(2t). Provide the full steps.

   Hint

   Start with the standard transform of sin(2t).

6. 6.

   Find the Laplace Transform of e^(-4t) (t² + t) and interpret the results given how the theorem modifies it.

   Hint

   Ensure you manage each component separately and apply the shift afterwards.