12.4. Properties of Inverse Laplace Transform

Try these first

1. 1.

   What does the linearity property express in the context of Laplace Transforms?

   Hint

   Remember that linearity relates to addition and scalar multiplication.

2. 2.

   State the time-shifting property in simple terms.

   Hint

   Think about how exponential terms can delay functions in time.

3. 3.

   What is the effect of the linearity property on Inverse Laplace Transforms?

   • It allows scalar multiplication
   • It allows function addition
   • Both A and B

   Hint

   Think about how we combine functions.

4. 4.

   Does time shifting affect how functions are presented?

   • True
   • False

   Hint

   Consider the example of delayed functions.

5. 5.

   Given the function F(s) = 5/(s^2 + 4), find the inverse transform using the shifts and decompositions that could apply.

   Hint

   Consider how to express constants in terms of known pairs.

6. 6.

   Consider a function F(s) = 2/(s - 1)^2. Use the properties to evaluate the inverse transform and discuss its implications.

   Hint

   What happens when you have repeated poles? Think about your functions' growth.