1.1.8. General Formula

Try these first

1. 1.

   What is the Laplace Transform of the function $f(t) = t$?

   Hint

   Use integration by parts to derive.

2. 2.

   Calculate $L\{f'''(t)\}$ for $f(t) = e^{3t}$.

   Hint

   Remember to apply the formula for the third derivative using initial values.

3. 3.

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

   • A) L{f'(t)} = sF(s) + f(0)
   • B) L{f'(t)} = sF(s) - f(0)
   • C) L{f'(t)} = F(s) - f(0)

   Hint

   Remember the negative sign for the initial value.

4. 4.

   True or False: The Laplace Transform can convert time-domain functions into frequency-domain functions.

   • True
   • False

   Hint

   Consider what the purpose of the Laplace Transform is.

5. 5.

   Given the initial value problem defined by the second-order linear homogeneous ODE $y'' - 3y' + 2y = 0$ with initial conditions $y(0) = 1$ and $y'(0) = 0$, find the general solution using the Laplace Transform.

   Hint

   Keep track of the initial values and re-arrange the equation.

6. 6.

   If $f(t) = t^3 e^{-2t}$, derive L{f^{(4)}(t)} without computing the individual derivatives.

   Hint

   Understand how to apply properties and series to achieve this without explicit derivative calculation.