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Math  /  Calculus

QuestionFind the Laplace transform of F(s)=□.f(t)=4u2(t)−4u4(t)−4u7(t)F(s)=\square . \quad f(t)=4 u_{2}(t)-4 u_{4}(t)-4 u_{7}(t)

Studdy Solution

STEP 1

What is this asking? We need to find the Laplace transform of a function f(t)f(t) that's built using unit step functions. Watch out! Don't forget to carefully handle the shifts introduced by the unit step functions!

STEP 2

1. Define the function
2. Apply the Laplace Transform

STEP 3

Alright, let's break down this function f(t)f(t)!
It's made up of several **unit step functions**, which essentially turn things "on" and "off" at specific times.
Think of them like switches!

STEP 4

Our function is f(t)=4u2(t)−4u4(t)−4u7(t)f(t) = 4 u_{2}(t) - 4 u_{4}(t) - 4 u_{7}(t).
This means we have a **4** turning on at t=2t=2, a **-4** turning on at t=4t=4, and another **-4** turning on at t=7t=7.

STEP 5

Now, let's apply the Laplace transform!
Remember the formula for the Laplace transform of a unit step function: L{uc(t)f(t−c)}=e−csF(s) \mathcal{L}\{u_c(t)f(t-c)\} = e^{-cs}F(s) where F(s)F(s) is the Laplace transform of f(t)f(t).

STEP 6

For our function, let's consider each term separately.
The Laplace transform of 4u2(t)4u_2(t) can be thought of as 4u2(t)⋅14u_2(t) \cdot 1, where the 11 represents a constant function that's always "on." Since L{1}=1s\mathcal{L}\{1\} = \frac{1}{s}, we have: L{4u2(t)}=4⋅e−2s⋅1s=4e−2ss \mathcal{L}\{4u_2(t)\} = 4 \cdot e^{-2s} \cdot \frac{1}{s} = \frac{4e^{-2s}}{s}

STEP 7

Similarly, for −4u4(t)-4u_4(t), we get: L{−4u4(t)}=−4⋅e−4s⋅1s=−4e−4ss \mathcal{L}\{-4u_4(t)\} = -4 \cdot e^{-4s} \cdot \frac{1}{s} = \frac{-4e^{-4s}}{s}

STEP 8

And for −4u7(t)-4u_7(t), we have: L{−4u7(t)}=−4⋅e−7s⋅1s=−4e−7ss \mathcal{L}\{-4u_7(t)\} = -4 \cdot e^{-7s} \cdot \frac{1}{s} = \frac{-4e^{-7s}}{s}

STEP 9

Finally, we put it all together by adding the individual Laplace transforms: L{f(t)}=4e−2ss−4e−4ss−4e−7ss \mathcal{L}\{f(t)\} = \frac{4e^{-2s}}{s} - \frac{4e^{-4s}}{s} - \frac{4e^{-7s}}{s}

STEP 10

The Laplace transform of f(t)f(t) is: F(s)=4e−2s−4e−4s−4e−7ss F(s) = \frac{4e^{-2s} - 4e^{-4s} - 4e^{-7s}}{s}

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