13.9. Examples

Try these first

1. 1.

   What is the convolution of f(t) = t and g(t) = e^{-t}?

   Hint

   Use integration by parts to solve the integral.

2. 2.

   How do we use the Laplace transform in solving differential equations?

   Hint

   Identify the initial conditions before applying the transform.

3. 3.

   What is the result of the convolution of f(t) = t and g(t) = e^{-t}?

   • (t-1) + e^{-t}
   • (t+1)e^{-t}
   • t^2 + e^{-t}

   Hint

   Think of the integral representation of convolution.

4. 4.

   True or False: Convolution allows us to convert a differential equation into an algebraic equation.

   • True
   • False

   Hint

   Consider the conversion process.

5. 5.

   Evaluate the convolution of f(t) = sin(t) and g(t) = cos(t). Provide detailed steps showing the setup and final result.

   Hint

   Use angles for simplifying the integration.

6. 6.

   Solve the differential equation y'' + 2y' + y = e^{-t} using convolution and provide the solution expression.

   Hint

   Express the right side with the impulse response for simplification.