9.2. Summary

Try these first

1. 1.

   What is the value of the unit step function $u(t-2)$ at $t=1$?

   Hint

   Remember that it equals 0 for times less than 'a'.

2. 2.

   State the general Laplace Transform formula for the unit step function.

   Hint

   Think about what parts relate to the shift and the function itself.

3. 3.

   What is the value of the unit step function $u(t-1)$ when $t < 1$?

   • 0
   • 1
   • Undefined

   Hint

   Think back to the definition of the unit step function.

4. 4.

   The Laplace Transform of the unit step function $u(t-a)$ is:

   • \\( \\frac{e^{-as}}{s} \\)
   • \\( \\frac{e^{as}}{s} \\)
   • \\( \\frac{1}{s} \\)

   Hint

   Use what you've learned about the exponential decay in transformation.

5. 5.

   Formulate the Laplace Transform for the function $e^{3t}u(t-2)$ and solve for its inverse.

   Hint

   Remember to adjust for the value of 'a' in the shifting.

6. 6.

   Solve $y'' + 4y = u(t-1)$ using the Laplace transform and find the complete solution.

   Hint

   Consider the initial conditions you have with this equation.