# 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 (1)
We apply the Laplace transform ,

where . Hence, ;

(2)
Inverse Laplace:

Since ,

and we get  [Differential Equations] [First Order D.E.] [Second Order D.E.]
[Geometry] [Algebra] [Trigonometry ]
[Calculus] [Complex Variables] [Matrix Algebra] S.O.S MATHematics home page

Do you need more help? Please post your question on our S.O.S. Mathematics CyberBoard. Author: Mohamed Amine Khamsi