THEOREM 7.4.1 Derivatives of Transforms If F(s) = L{f(t)} and n = 1, 2, 3, .. . , then L{ R} = (-1)" F(s). ds Evaluate the given Laplace transform. (Write your answer as a function of s.) %3! Lte" sin(4t)} [(s-2)-10?
THEOREM 7.4.1 Derivatives of Transforms If F(s) = L{f(t)} and n = 1, 2, 3, .. . , then L{ R} = (-1)" F(s). ds Evaluate the given Laplace transform. (Write your answer as a function of s.) %3! Lte" sin(4t)} [(s-2)-10?
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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![**Theorem 7.4.1: Derivatives of Transforms**
If \( F(s) = \mathcal{L}\{f(t)\} \) and \( n = 1, 2, 3, \ldots \), then
\[
\mathcal{L}\{t^n f(t)\} = (-1)^n \frac{d^n}{ds^n} F(s).
\]
**Evaluate the given Laplace transform. (Write your answer as a function of \( s \).)**
\[
\mathcal{L}\{t e^{2t} \sin(4t)\}
\]
The result shown is:
\[
\frac{8}{[(s-2)^2 + 16]^2}
\]
This formula shows how to evaluate the Laplace transform of a function involving \( t \), exponential, and sine components using the theorem of derivatives of transforms. The boxed result is the solution expressed in terms of \( s \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9bf7f6f8-9470-47f0-a3a4-f3c052528216%2F6f5a462e-2082-49fc-abe0-e996d6179a6d%2Fcwn9ug_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Theorem 7.4.1: Derivatives of Transforms**
If \( F(s) = \mathcal{L}\{f(t)\} \) and \( n = 1, 2, 3, \ldots \), then
\[
\mathcal{L}\{t^n f(t)\} = (-1)^n \frac{d^n}{ds^n} F(s).
\]
**Evaluate the given Laplace transform. (Write your answer as a function of \( s \).)**
\[
\mathcal{L}\{t e^{2t} \sin(4t)\}
\]
The result shown is:
\[
\frac{8}{[(s-2)^2 + 16]^2}
\]
This formula shows how to evaluate the Laplace transform of a function involving \( t \), exponential, and sine components using the theorem of derivatives of transforms. The boxed result is the solution expressed in terms of \( s \).
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