15.7. Properties of Laplace Transforms

Try these first

1. 1.

   What does the linearity property of the Laplace transform state?

   Hint

   Think about breaking down functions.

2. 2.

   What is the formula for the First Shifting Theorem?

   Hint

   Remember the shift around 's'!

3. 3.

   What is the formula for the linearity property of the Laplace transform?

   • L{af(t)} + L{bg(t)}
   • aL{f(t)} + bL{g(t)}
   • a + b

   Hint

   Focus on how you can distribute the transform.

4. 4.

   True or False: The First Shifting Theorem states that L{e^{it}f(t)} = F(s+i).

   • True
   • False

   Hint

   Remember how the shift occurs.

5. 5.

   Using the properties of Laplace transforms, solve the differential equation y'' + 3y' + 2y = Heaviside(t-2), where Heaviside is a unit step function.

   Hint

   Use the properties to separate the functions effectively.

6. 6.

   Demonstrate solving a second-order ODE using the First Shifting Theorem applied to the function e^{5t}sin(t).

   Hint

   Identify how the shift impacts the s variable.