On this page, we discuss the second of two important theorems related to Laplace Transforms. They are rather cleverly named the First Shifting Theorem and the Second Shifting Theorem.
\(\displaystyle{ \mathcal{L}\{ e^{-at} f(t) \} = F(s+a) }\) | |
Second Shifting Theorem |
\(\displaystyle{ \mathcal{L}\{ u(t-c)g(t-c) \} = e^{-cs}G(s) }\) |
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Second Shifting Theorem
The second shifting theorem looks similar to the first but the results are quite different.   In the \(t\)-domain we have the unit step function (Heaviside function) which translates to the exponential function in the \(s\)-domain.   Your Laplace Transforms table probably has a row that looks like \(\displaystyle{ \mathcal{L}\{ u(t-c)g(t-c) \} = e^{-cs}G(s) }\)
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 |
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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 |
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Okay, use the second shifting theorem to solve these problems.
Practice
Unless otherwise instructed,
- if \(f(t)\) is given, find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the second 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.
\( f(t) = (t+1)^2 e^t \)
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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\( f(t) = e^{2t} \cos(3t) \)
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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\(\displaystyle{ F(s) = \frac{s-2}{s^2-4s+5} }\)
Problem Statement
For \(\displaystyle{ F(s) = \frac{s-2}{s^2-4s+5} }\), 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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\(\displaystyle{ F(s) = \frac{1}{s^2-4s+5} }\)
Problem Statement
For \(\displaystyle{ F(s) = \frac{1}{s^2-4s+5} }\), 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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Practice Instructions
Unless otherwise instructed,
- if \(f(t)\) is given, find the Laplace transform \( F(s) = \mathcal{L}\{ f(t) \} \) using the second 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.