8.1.3. Proof of the Division by t Rule

Try these first

1. 1.

   What is the Division by t rule?

   Hint

   Think about how division relates to integration.

2. 2.

   Which theorem allows us to switch the order of integration in proving the Division by t rule?

   Hint

   Recall that it deals with iterated integrals.

3. 3.

   What is the relationship between division by t in the Laplace domain?

   • Multiplication by s
   • Integration
   • Differentiation

   Hint

   Consider how operations in one domain connect to those in another.

4. 4.

   True or False: The Division by t rule can be used for any function.

   • True
   • False

   Hint

   Think about the requirements for applying the rule.

5. 5.

   Prove the Division by t Rule for the function $e^{at}$.

   Hint

   Start with the definition of the Laplace transform.

6. 6.

   Apply the Division by t Rule to find the Laplace transform of a more complex function, for example, $\frac{e^{-at}}{t}$.

   Hint

   Use known transforms to simplify the integration process.