18.2.3. Step 3: Apply the inverse Laplace Transform to find 𝑓(𝑡)

Try these first

1. 1.

   What is the purpose of the inverse Laplace Transform?

   Hint

   Think about what we want to achieve with the function after using Laplace Transform.

2. 2.

   Define a Volterra Integral Equation.

   Hint

   Consider the form of integral equations we discussed.

3. 3.

   What does the inverse Laplace Transform retrieve?

   • Algebraic equation
   • Time-domain function
   • Integral equation

   Hint

   Consider what you ultimately want to find in function analysis.

4. 4.

   True or False: The Laplace Transform can simplify integral equations to algebraic equations.

   • True
   • False

   Hint

   Recall how we use Laplace in engineering applications.

5. 5.

   Given 𝐹(𝑠) = 1 / (s² + 9), find the inverse Laplace Transform and express 𝑓(𝑡).

   Hint

   Look for trigonometric forms in your inverse Laplace Transform tables.

6. 6.

   Consider the equation 𝑓(𝑡) = 1 + ∫ (t - τ)f(τ)dτ. Solve for 𝑓(𝑡) by following the steps of Laplace and inverse.

   Hint

   Follow the structured steps of the Laplace Transform to derive the equation and solve.