13.7. Solving Differential Equations Using Convolution

Try these first

1. 1.

   Define convolution in your own words.

   Hint

   Think about how one function modifies another.

2. 2.

   What does the Laplace Transform do?

   Hint

   It simplifies solving differential equations.

3. 3.

   What is the primary purpose of using convolution in differential equations?

   • To multiply functions directly
   • To transform them into a solvable form
   • To simplify integration

   Hint

   Think about how we switch from differential to algebraic equations.

4. 4.

   True or False: The impulse response function tells us how the system responds to a step input.

   • True
   • False

   Hint

   Recall the definition of impulse in systems theory.

5. 5.

   Given the equation $y'' + y = e^{-t}$, solve for $y(t)$ using convolution.

   Hint

   Remember to apply initial conditions properly.

6. 6.

   If $f(t) = e^{-2t}$ and $h(t) = t e^{-t}$, compute $(f * h)(t)$.

   Hint

   Make sure to keep track of the limits properly while integrating.