1.1. Laplace Transform of Derivatives

Try these first

1. 1.

   What is the formula for L{f'(t)}?

   Hint

   Think about the first derivative and its relationship to the Laplace Transform.

2. 2.

   What initial condition is used in L{f'(t)}?

   Hint

   Recall how initial conditions affect derivatives.

3. 3.

   What is L{f'(t)}?

   • sF(s) - f(0)
   • s^2F(s) - sf(0) - f'(0)
   • F(s)

   Hint

   Think about how derivatives are transformed.

4. 4.

   True or False: The Laplace Transform can be used to solve ODEs.

   • True
   • False

   Hint

   Consider the application of Laplace in mathematics and engineering.

5. 5.

   Consider the initial value problem y'' + 4y = 0 where y(0)=1 and y'(0)=0. Solve using the Laplace Transform.

   Hint

   Pay attention to the transformation properties and how to apply initial conditions.

6. 6.

   Use the Laplace Transform to find the solution for the differential equation y'' - 3y' + 2y = e^{2t} with initial conditions y(0)=0 and y'(0)=1.

   Hint

   Take advantage of known transforms for e^at and its derivatives.