# Dirac delta function

### Dirac delta function

#### Lessons

The Dirac Delta function can be thought of as an instantaneous impulse

There are 3 main conditions for the Dirac Delta function:

1.

$\delta$($t - c$) = $\infty$, $t = c$

2.

$\delta$($t - c$) = 0, $t$ $\neq$ $c$

3.

$\int_{-\infty}^{\infty}\delta(t - c)dt = 1, \epsilon$ > 0

The Laplace Transform of a Dirac Delta Function is:

$L${$\delta$($t - c$)} = $e^{-sc}$, provided $c$ > 0

We can also relate the Dirac Delta Function to the Heaviside Step Function:

$u'_{c}(t) = \delta(t - c)$

• Introduction
a)
What is a Dirac Delta Function? And what is the Laplace Transform of a Dirac Delta Function?

b)
Relating the Dirac Delta Function to the Heaviside Step Function

• 1.
Calculating the Laplace Transform of Dirac Delta Functions

Solve the following equations:

a)
$u_{4}(t)\delta(t - 3)$

b)
$L${3$\delta$($t$ - 7)}

• 2.
Solving Differential Equations with Dirac Delta Functions

Solve the following differential equation,

$y'' - 3y' + 2y = 2\delta(t - 3)$

Where $y$(0) = 1, $y$'(0) = 3