1.1.6. Proof of Second Derivative

Try these first

1. 1.

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

   Hint

   Recall how derivatives are transformed.

2. 2.

   Write down the definition of exponential order.

   Hint

   Think about how functions behave at infinity.

3. 3.

   What does the Laplace Transform of the first derivative give you?

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

   Hint

   Think back to the formulaapplied to f(t) and it's derivative.

4. 4.

   True or False: The Laplace Transform can be used for higher-order derivatives.

   • True
   • False

   Hint

   Recall the generalization discussed for n-th derivatives.

5. 5.

   Given the function f(t) = e^{2t}, compute L{f''(t)} and verify the initial conditions.

   Hint

   Apply the basic rules of Laplace for differentiating functions.

6. 6.

   Create and solve an initial value problem for a second-order linear ODE using the Laplace Transform.

   Hint

   Use the formula for L{y''(t)} in your setup.