1.1.7. Laplace Transform of the n-th Derivative

Try these first

1. 1.

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

   Hint

   Remember the structure involves s and the initial function value.

2. 2.

   True or False: The Laplace Transform can only be applied to continuous functions.

   Hint

   Consider the nature of functions required for transformation.

3. 3.

   What does L{f′(t)} equal?

   • sF(s) - f(0)
   • sF(s) + f(0)
   • F(s) - s

   Hint

   Think about how derivatives reduce to algebra.

4. 4.

   True or False: The Laplace Transform can be used for integral equations.

   • True
   • False

   Hint

   Recall the scope of the transform discussed.

5. 5.

   Find the Laplace Transform of f(t) = e^3t cos(4t), using the property L{e^{at} f(t)} = F(s-a).

   Hint

   Identify a and b from the exponential and cosine components.

6. 6.

   Solve the differential equation y'' + 4y = 0 with y(0)=1 and y'(0)=0 using the Laplace Transform.

   Hint

   Transform each term carefully while applying initial conditions.