7.1. Multiplication by tn (Power of t)

Try these first

1. 1.

   What is the Laplace Transform of $e^{at}$?

   Hint

   Recall the basic property of Laplace Transforms.

2. 2.

   State the multiplication property when multiplying by $tn$.

   Hint

   Think about how differentiation affects the transform.

3. 3.

   What does the multiplication by $tn$ property do to a function's Laplace Transform?

   • It massifies the function.
   • It shifts the function.
   • It differentiates the transform \\(n\\) times with alternating sign.

   Hint

   Focus on how multiplication affects differentiation.

4. 4.

   True or False: The function must be continuous everywhere for the multiplication by $tn$ property to hold.

   • True
   • False

   Hint

   Remember the conditions for the Laplace Transform.

5. 5.

   Prove the general multiplication property for $n=2$ using integration by parts.

   Hint

   Use the integration by parts technique for differentiation.

6. 6.

   Given $f(t) = e^{2t}$, find $L\{t^3 f(t)\}$ and show all steps.

   Hint

   Compute \\(L\\{e^{2t}\\}\\) first to establish a solid base for your derivatives.