12.6. Practice Problems

Try these first

1. 1.

   Find the inverse Laplace transform of L⁻¹{1 / s^2}.

   Hint

   Recall that L⁻¹{1/s^n} corresponds to t^(n-1)/(n-1)!

2. 2.

   What is L⁻¹{5 / s + 2}?

   Hint

   Use the standard inverse transform for exponential decay.

3. 3.

   What does the Inverse Laplace Transform do?

   Hint

   Think about the relationship between time and frequency functions.

4. 4.

   Partial fraction decomposition is used for what purpose?

   • To simplify fractions
   • To multiply functions
   • To take derivatives

   Hint

   It's about breaking down functions to find their inverses.

5. 5.

   Evaluate L⁻¹{(s² + 5s + 6) / (s³ + 4s² + 5s)}. Show all steps in the solution.

   Hint

   Factor the denominator first, and then apply partial fractions.

6. 6.

   Use the Convolution Theorem to find L⁻¹{(s+3)(s+2)/(s² + 4)(s + 1)}. Explain each step.

   Hint

   Identify the two parts to convolve and apply the integral accordingly.