Dirac delta function

Dirac delta function

Lessons

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

Dirac Delta

There are 3 main conditions for the Dirac Delta function:

1.

δ\delta(tct - c) = \infty, t=ct = c

2.

δ\delta(tct - c) = 0, tt \neq cc

3.

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

The Laplace Transform of a Dirac Delta Function is:

LL{δ\delta(tct - c)} = esce^{-sc}, provided cc > 0

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

uc(t)=δ(tc)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)
    u4(t)δ(t3)u_{4}(t)\delta(t - 3)

    b)
    LL{3δ\delta(tt - 7)}


  • 2.
    Solving Differential Equations with Dirac Delta Functions

    Solve the following differential equation,

    y3y+2y=2δ(t3)y'' - 3y' + 2y = 2\delta(t - 3)

    Where yy(0) = 1, yy'(0) = 3