1.5. Examples

Try these first

1. 1.

   What is the Laplace Transform of f(t) = 1?

   Hint

   Think about how a constant behaves under the transform.

2. 2.

   What is the condition for the existence of the Laplace Transform of a function?

   Hint

   Consider the characteristics of the function.

3. 3.

   What is ℒ{1}?

   • 1/s
   • s
   • 1

   Hint

   Consider what the transform of a constant means mathematically.

4. 4.

   True or False: The Laplace Transform of e^(3t) is valid only when s > 3.

   • True
   • False

   Hint

   Reflect on how rapid growth affects convergence.

5. 5.

   Analyze the function f(t) = e^(2t) + 5 and find its Laplace Transform. Justify the conditions required for the transform.

   Hint

   Consider how linearity helps in combining transforms for multiple parts of a function.

6. 6.

   Given the function f(t) = 3t^4 - 2e^{t}, compute its Laplace Transform and discuss the convergence conditions.

   Hint

   Separate the terms and apply the Laplace Transform to each part individually.