1. Laplace Transforms & Applications

Try these first

1. 1.

   What is the Laplace Transform of a constant function, L{1}?

   Hint

   Consider how the Laplace Transform deals with constants.

2. 2.

   Write down the formula for L{f′(t)}.

   Hint

   Think about the first derivative in the Laplace domain.

3. 3.

   What is the formula for the Laplace Transform of the first derivative?

   • L{f′(t)} = sF(s) - f(0)
   • L{f′(t)} = sF(s) + f(0)
   • L{f′(t)} = F(s) - sf(0)

   Hint

   Focus on the relationship between derivatives and algebraic transformations.

4. 4.

   True or False: The Laplace Transform can only be applied to linear differential equations.

   • True
   • False

   Hint

   Remember the nature of types of differential equations.

5. 5.

   Using Laplace Transforms, derive the solution for the system y'' + 4y' + 4y = 0, y(0) = 1, y'(0) = 0.

   Hint

   Apply the transformations step by step for clarity.

6. 6.

   Demonstrate the use of Laplace Transforms in finding L{e^at sin(bt)}.

   Hint

   Plug into the transformation formulas wisely.