Impulse Functions: Dirac Function

It is very common for physical problems to have impulse behavior, large quantities acting over very short periods of time. These kinds of problems often lead to differential equations where the nonhomogeneous term g(t) is very large over a small interval  and is zero otherwise. The total impulse of g(t) is defined by the integral

In particular, let us assume that g(t) is given by

where the constant  is small. It is easy to see that  . When the constant  becomes very small the value of the integral will not change. In other words,
,
while

This will help us define the so-called Dirac delta-function by

If we put  , then we have

More generally, we have


Example: Find the solution of the IVP

Solution. We follow these steps:

(1)

We apply the Laplace transform,
where  . Hence,
;

(2)

Inverse Laplace:Since
,
and

we get