18.3. Concept Overview: Laplace Transform of Derivatives

Try these first

1. 1.

   What is the Laplace transform of the function f(t) = 1?

   Hint

   Use the definition of Laplace transforms.

2. 2.

   What is the first derivative formula in the Laplace transform?

   Hint

   Recall that it involves the initial value.

3. 3.

   What is the formula for the first derivative in Laplace transforms?

   • L{d²f(t)/dt²} = s²F(s)
   • L{df(t)/dt} = sF(s) - f(0)
   • L{d³f(t)/dt³} = s³F(s)

   Hint

   Focus on the first-order derivative.

4. 4.

   True or False: The Laplace transform converts initial conditions into algebraic formats.

   • True
   • False

   Hint

   Think about how initial conditions are integrated.

5. 5.

   Using Laplace transforms, solve the ODE d³y/dt³ + 5d²y/dt² + 6dy/dt = 0 with initial conditions y(0)=2, y'(0)=1, y''(0)=0.

   Hint

   Factoring the polynomial in the s-domain will give you a better insight.

6. 6.

   Determine the Laplace transform of f(t) = t^2e^(-3t) and find its first derivative using this approach.

   Hint

   You may need to use differentiation under the integral sign.