10.4. Examples

ℒ{𝛿(𝑡−3)} = 𝑒^{−3𝑠}

In this example, we take the Laplace Transform of the Dirac Delta Function, specifically the shifted version δ(t - 3). According to the property of Laplace Transforms for the delta function, this translates to an exponential decay term. The '3' indicates that the impulse occurs at t = 3, and thus the result becomes e^{-3s}, showing how the time shift alters the Laplace domain representation.

Imagine you are at a concert, and a sudden loud sound occurs at exactly 3 minutes into the show. That moment can be depicted as an impulse at time t = 3, and the Laplace Transform showcases how this sharp sound is represented in a mathematical form, showing how it would influence the overall audio system's response.

ℒ{5𝛿(𝑡−2)} = 5𝑒^{−2𝑠}

In this case, we are looking at 5 times the Dirac Delta Function shifted to t = 2. Here, the Laplace Transform results in 5e^{-2s}. The factor of 5 indicates that the impulse is magnified, meaning rather than a single instantaneous effect, the impulse is five times stronger. This illustrates how scaling the delta function affects its Laplace representation.

Think of this as a more intense musical hit at 2 seconds during a song, like a drummer hitting a cymbal harder than usual. The magnitude of the sound spike (5 times louder) impacts how the sound system responds, and the Laplace Transform encapsulates all of this energy in a simple mathematical form.

Given:

d𝑦/d𝑡 + 𝑦 = 𝛿(𝑡−2), 𝑦(0) = 0

Taking Laplace on both sides:

e^{−2𝑠}

𝑠𝑌(𝑠)+𝑌(𝑠) = 𝑒^{−2𝑠} ⇒ 𝑌(𝑠) = e^{−2𝑠}/(𝑠+1)

Taking Inverse Laplace:

y(𝑡) = ℒ^{−1}{(e^{−2𝑠}/(𝑠+1))} = 𝑢(𝑡−2)e^{−(𝑡−2)}

This example shows how to solve a differential equation that includes an impulse input represented by δ(t - 2). By applying the Laplace Transform, we convert the equation into an algebraic form which is more straightforward to solve. The final result reveals that the solution y(t) is influenced by the Heaviside step function u(t - 2), indicating that the response begins at t = 2 and then decays exponentially after that point.

Consider a scenario where a fire alarm is triggered at 2 minutes into a movie. The alarm can be modeled as an impulse function causing a sudden response—initially, there is no sound (y(0) = 0), but when the alarm rings (the impulse input), it influences everyone in the vicinity, and their response can be modeled as a delay followed by an exponential decrease in noise as they calm down, just like the solution derived from this differential equation.

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