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

**
**

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