Find the laplace transform of the following function f(t) = 58(t-7)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
Find the Laplace transform of the following function:
\[ f(t) = 5\delta(t - 7) \]
**Explanation:**
In this problem, we are tasked with finding the Laplace transform of a scaled and shifted Dirac delta function. The function \( f(t) = 5\delta(t - 7) \) represents a Dirac delta function that is scaled by a factor of 5 and shifted to the right by 7 units along the time axis.
The Dirac delta function, \( \delta(t - a) \), is a mathematical construct that is zero everywhere except at \( t = a \), where it is infinitely high such that its integral over time is one. It is often used to model impulsive forces or signals in systems.
The Laplace transform of a Dirac delta function \( \delta(t - a) \) is given by:
\[ \mathcal{L}\{\delta(t - a)\} = e^{-as} \]
Therefore, using the property of linearity in Laplace transforms, the Laplace transform of the given function is:
\[ \mathcal{L}\{f(t)\} = 5e^{-7s} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3ba71b9f-6645-4b82-be5d-cda1c9d5ec57%2Fa85be4c4-cd63-4439-8fb7-1c75811cc228%2Fd7j7cp5_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the Laplace transform of the following function:
\[ f(t) = 5\delta(t - 7) \]
**Explanation:**
In this problem, we are tasked with finding the Laplace transform of a scaled and shifted Dirac delta function. The function \( f(t) = 5\delta(t - 7) \) represents a Dirac delta function that is scaled by a factor of 5 and shifted to the right by 7 units along the time axis.
The Dirac delta function, \( \delta(t - a) \), is a mathematical construct that is zero everywhere except at \( t = a \), where it is infinitely high such that its integral over time is one. It is often used to model impulsive forces or signals in systems.
The Laplace transform of a Dirac delta function \( \delta(t - a) \) is given by:
\[ \mathcal{L}\{\delta(t - a)\} = e^{-as} \]
Therefore, using the property of linearity in Laplace transforms, the Laplace transform of the given function is:
\[ \mathcal{L}\{f(t)\} = 5e^{-7s} \]
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