Lesson 27 - The Dirac Delta Function

Here, we explore time shift equations a bit more by introducing the Dirac delta function. $\delta (t-a)$.

1. The Dirac delta function $\delta (t-a)$ is characterized by the following two properties:

a)

b)

for any continuous function $f(t)$ on an open interval containing $t=0$.

2. $\mathcal{L}[\delta (t-a)]=e^{-as}$, ${\mathcal{L}}^{-1}[e^{-as}]=\delta (t-a)$

3. Reminder: ${\mathcal{L}}^{-1}[e^{-as}F(s)]=u(t-a)f(t-a)$

4. We can use: ${\mathcal{L}}^{-1}\left[\frac{a}{s^2-a^2}\right]=\sinh at$, ${\mathcal{L}}^{-1}\left[\frac{s}{s^2-a^2}\right]=\cosh at$

Examples

Find: $\mathcal{L}[\delta (t-1)-\delta (t-3)]$

Find: $\mathcal{L}[e^t\delta (t-3)]$

Here, we take the Laplace transform of the $\delta (t-3)$ function. However, now that we have an $e^t$, we perform a shift based on the term $e^t$.

First, we will take the Laplace transform of the Dirac delta function.

Now, we will apply the time shift. The time shift is in the form $e^{at}$. Here, $a=1$, so we will shift the opposite of the sign of $a$, so a negative shift.

So,

Let's try another.

Find: $\mathcal{L}[e^{-3t}\delta (t-5)]$

Again, we take the Laplace transform of $\delta (t-5)$, this gives us $e^{-5s}$. Then we shift by the $a$ in $e^{at}$. This gives us $3^{-5(s+3)}$.

Try For Yourself

1. Find: $\mathcal{L}[e^{2t}\delta (t+2)]$
2. Find: $\mathcal{L}[e^{-t}\delta (t-3)]$
3. Find: $\mathcal{L}[e^{-2t}\delta (t-1)]$

We can also take the inverse Laplace transform of a Dirac delta function. We typically find the inverse Laplace transform of $F(s)$ as we normally do, then we time shift from the time shift term. Let's look at an example.

Example

${\mathcal{L}}^{-1}\left[\frac{1}{s{s}^2+3s+2}{e}^{-2s}\right]$

First, identify $F(s)$.

Now, we take the inverse Laplace transform of $F(s)$.

Partial fraction decomposition:

So,

Now,

We know the Laplace transforms of $\frac{1}{s+1}$ and $\frac{1}{s+2}$.

Now, we apply the inverse Laplace transform of the time shift $e^{-2s}$.

Try for Yourself

1. Find: ${\mathcal{L}}^{-1}\left[\frac{1}{s^2+2s+5}{e}^{-2s}\right]$

On the page, you will find additional practice problems including an application problem regarding Dirac delta functions.

Next Lesson: Lesson 27.1 - Extra Practice Solving IVPs Containing Dirac Delta Functions

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