Math Problem Statement
Find the Laplace Transform of 𝑓(𝑡)={3𝑡 , 0< 𝑥<27 , 2<𝑥<4
Solution
It seems there are a few notational issues in your question. I assume that you are trying to find the Laplace Transform of a piecewise function defined as:
\begin{cases} 3t & 0 < t < 2 \\ 7 & 2 < t < 4 \\ \end{cases}$$ To solve for the Laplace transform of this function, we can break it into two parts, because it is defined in pieces over different intervals. The Laplace transform $$\mathcal{L}$$ of a function is given by: $$\mathcal{L}\{f(t)\} = \int_{0}^{\infty} e^{-st} f(t) \, dt$$ Given that the function is piecewise, we will compute the Laplace transform over the two intervals separately: ### First Interval: $$0 < t < 2$$, where $$f(t) = 3t$$ The Laplace transform of $$3t$$ from $$0$$ to $$2$$ is: $$\mathcal{L}\{3t\} = \int_0^2 e^{-st} 3t \, dt$$ We can solve this integral using integration by parts. ### Second Interval: $$2 < t < 4$$, where $$f(t) = 7$$ The Laplace transform of a constant function (7) over the interval from 2 to 4 is: $$\mathcal{L}\{7\} = \int_2^4 e^{-st} 7 \, dt$$ This is a simple exponential integral. Let me know if you'd like the full solutions and step-by-step breakdown for these integrals!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Laplace Transform
Piecewise Functions
Integration by Parts
Exponential Integrals
Formulas
Laplace Transform: L{f(t)} = ∫₀^∞ e^(-st) f(t) dt
Theorems
Integration by Parts
Laplace Transform of Piecewise Functions
Suitable Grade Level
Grades 11-12 / Undergraduate
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