18.1. Laplace Transform Approach

Try these first

1. 1.

   What is an integral equation?

   Hint

   Think of the definition.

2. 2.

   What is the role of the kernel in a Volterra equation?

   Hint

   Focus on its effect on the unknown function.

3. 3.

   What is the general form of a Volterra integral equation of the second kind?

   Hint

   Remember the terms in the equation.

4. 4.

   The Laplace Transform simplifies the solution of integral equations through which theorem?

   • Integration Theorem
   • Convolution Theorem
   • Transform Theorem

   Hint

   Think about integral properties.

5. 5.

   Given the integral equation f(t) = 3 + ∫_0^t (t - τ)f(τ) dτ, solve for f(t) using the Laplace Transform.

   Hint

   Apply the Laplace Transform and manipulate the equation.

6. 6.

   For f(t) = sin(t) + ∫_0^t (t - τ)f(τ) dτ, determine and provide f(t).

   Hint

   Think about how Laplace Transform of sin(t) helps you solve this.