8.1.5.1. Example 2

Try these first

1. 1.

   What is the division by t rule in Laplace transforms?

   Hint

   Think about what integration means in the s-domain.

2. 2.

   Name one application of the division by t property.

   Hint

   Consider fields where time-varying inputs are encountered.

3. 3.

   What does the division by t rule correspond to in the s-domain?

   • Multiplication
   • Integration
   • Differentiation

   Hint

   Link division operations with their respective integration implications.

4. 4.

   The division by t rule is applicable only when the function is:

   • True
   • False

   Hint

   Consider the continuity conditions for applying the transformations.

5. 5.

   Using the division by t rule, find the Laplace transform of a piecewise function defined by $f(t) = \sin(at), t \leq 1$ and $1, t > 1$.

   Hint

   Break the function into manageable parts before applying the rule.

6. 6.

   Demonstrate how the division by t rule can aid in resolving the differential equation $y'' + 2y' + y = \frac{1}{t}$ with initial conditions.

   Hint

   Analyze the terms separately and employ Laplace properties effectively.