1.1.3. Laplace Transform of the First Derivative

Try these first

1. 1.

   What is the Laplace Transform of the function f(t) = t?

   Hint

   Use the formula for the Laplace Transform.

2. 2.

   Write down the formula for the Laplace Transform of the first derivative.

   Hint

   Remember the connection between `F(s)` and `f(0)`.

3. 3.

   What is the Laplace Transform of the first derivative?

   Hint

   Recall the derived formula from class.

4. 4.

   Laplace Transform converts which type of equations into algebraic ones?

   • True
   • False

   Hint

   Think about how we convert differential terms into algebraic ones.

5. 5.

   Derive the Laplace Transform of f(t) = e^{at} and show how to apply it to a differential equation.

   Hint

   Start from basic properties of the Laplace Transform.

6. 6.

   Given y' + 2y = e^{3t} with y(0) = 1, use Laplace Transform to find y(t).

   Hint

   Remember to apply the Laplace of the forcing term correctly.