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

where

,

is called **the convolution product** of *f*(*t*), and
*g*(*t*). The following are some of
the basic properties of the convolution product:

**(1)**- ;
**(2)**- ;
**(3)**- ;
**(4)**- ;
**(5)**- .

**Example:** Find the solution to

**Solution:** Apply the Laplace transform to get

where . Hence,

We then rewrite *Y*(*s*) to get

Therefore, we have

**
**

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