One of the most common mistake students will commit is
Although it is tempting to assume that this is true, one may easily check that it is wrong by taking f(t) = t, and g(t) = t. Therefore, a natural question is: what would be equal to ?
The answer to this question is given by the formula
is called the convolution product of f(t), and g(t). The following are some of the basic properties of the convolution product:
Solution: Apply the Laplace transform to get
where . Hence,
We then rewrite Y(s) to get
Therefore, we have