On this page, we discuss two important theorems related to Laplace Transforms. They are rather cleverly named the First Shifting Theorem and the Second Shifting Theorem.
\(\displaystyle{ \mathcal{L}\{ e^{at} f(t) \} = F(s+a) }\)  
\(\displaystyle{ \mathcal{L}\{ u(tc)g(tc) \} = e^{cs}G(s) }\) 
First Shifting Theorem
The first shifting theorem says that in the tdomain, if we multiply a function by \(e^{at}\), this results in a shift in the sdomain a units. In your Laplace Transforms table you probably see the line that looks like
\(\displaystyle{ \mathcal{L}\{ e^{at} f(t) \} = F(s+a) }\)
This idea looks easy and watching other people using it may look easy but when you have to do it, it is not so easy. So it is important to get your hands dirty by working some problems on your own. After you watch this first video, you can work some practice problems to help you understand this better.
Let's watch a video that explains how this works. This video contains a very good explanation, a couple of examples and the proof of this theorem.
video by Dr Chris Tisdell 

Practice  First Shifting Theorem
Use the First Shifting Theorem on these practice problems.
Unless otherwise instructed,
 if \(f(t)\) is given, find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem
 if \(F(s)\) is given, find the inverse Laplace transform \( f(t) = \mathcal{L}^{1} \{ F(s) \} \)
Give your answers in exact, completely factored form.
Basic
\( f(t) = e^{4t}t^6 \)
Problem Statement 

For \( f(t) = e^{4t}t^6 \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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\( f(t) = (t+1)^2 e^t \)
Problem Statement 

For \( f(t) = (t+1)^2 e^t \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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\( f(t) = e^{3t} t^2 \)
Problem Statement 

For \( f(t) = e^{3t} t^2 \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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\( f(t) = e^t \sin(t) \)
Problem Statement 

For \( f(t) = e^t \sin(t) \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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\(\displaystyle{ F(s) = \frac{1}{(s+1)^2} }\)
Problem Statement 

For \(\displaystyle{ F(s) = \frac{1}{(s+1)^2} }\), find the inverse Laplace transform \( f(t) = \mathcal{L}^{1} \{ F(s) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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\(\displaystyle{ F(s) = \frac{\pi}{(s+\pi)^2} }\)
Problem Statement 

For \(\displaystyle{ F(s) = \frac{\pi}{(s+\pi)^2} }\), find the inverse Laplace transform \( f(t) = \mathcal{L}^{1} \{ F(s) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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Intermediate
\( f(t) = e^{2t} \cos(3t) \)
Problem Statement 

For \( f(t) = e^{2t} \cos(3t) \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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\(\displaystyle{ F(s) = \frac{s2}{s^24s+5} }\)
Problem Statement 

For \(\displaystyle{ F(s) = \frac{s2}{s^24s+5} }\), find the inverse Laplace transform \( f(t) = \mathcal{L}^{1} \{ F(s) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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\(\displaystyle{ F(s) = \frac{1}{s^24s+5} }\)
Problem Statement 

For \(\displaystyle{ F(s) = \frac{1}{s^24s+5} }\), find the inverse Laplace transform \( f(t) = \mathcal{L}^{1} \{ F(s) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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Second Shifting Theorem
The second shifting theorem looks similar to the first but the results are quite different. In the tdomain we have the unit step function (Heaviside function) which translates to the exponential function in the sdomain. Your Laplace Transforms table probably has a row that looks like \(\displaystyle{ \mathcal{L}\{ u(tc)g(tc) \} = e^{cs}G(s) }\)
Okay, let's watch a video that explains this very well and contains a couple of examples and discusses the big picture.
video by Dr Chris Tisdell 

This next video clip contains a lot of the same explanation as the previous video but, toward the end, shows why the theorem holds. It is a good video to watch after the previous one to get some repetition and more detailed explanation. Then, work some practice problems to hone your skills.
video by Dr Chris Tisdell 

Practice  Second Shifting Theorem
Use the Second Shifting Theorem on these practice problems.
Unless otherwise instructed,
 if \(f(t)\) is given, find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem
 if \(F(s)\) is given, find the inverse Laplace transform \( f(t) = \mathcal{L}^{1} \{ F(s) \} \)
Give your answers in exact, completely factored form.
Basic
\( f(t) = u(t4)[t4] \)
Problem Statement 

For \( f(t) = u(t4)[t4] \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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\( f(t) = u(t4)[t4]^2 \)
Problem Statement 

For \( f(t) = u(t4)[t4]^2 \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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\( f(t) = u(t2\pi) \sin(t2\pi) \)
Problem Statement 

For \( f(t) = u(t2\pi) \sin(t2\pi) \), find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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\(\displaystyle{ F(s) = \frac{1e^{2s}}{s^2} }\)
Problem Statement 

For \(\displaystyle{ F(s) = \frac{1e^{2s}}{s^2} }\), find the inverse Laplace transform \( f(t) = \mathcal{L}^{1} \{ F(s) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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Intermediate
\( f(t) = \cos[3(t1)] u(t1) \)
Problem Statement 

For \( f(t) = \cos[3(t1)] u(t1) \) find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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\( f(t) = \left\{\begin{array}{lr} t & 0 \leq 4 \\ 2 & t = 4 \\ 0 & t > 4 \end{array} \right. \)
Problem Statement 

For \( f(t) = \left\{\begin{array}{lr} t & 0 \leq 4 \\ 2 & t = 4 \\ 0 & t > 4 \end{array} \right. \) find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem.
Solution 

video by Dr Chris Tisdell 

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Practice Instructions
Unless otherwise instructed,
 if \(f(t)\) is given, find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using a shifting theorem
 if \(F(s)\) is given, find the inverse Laplace transform \( f(t) = \mathcal{L}^{1} \{ F(s) \} \)
Give your answers in exact, completely factored form.